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LANCELOT HOGBEN 



Science for the Citizen 

A Self-Educator based on the Social 
Background of Scientific Discovery 



"Such philosophy as shall not vanish in the fume 
of subtile, sublime, or delectable speculation but 
shall be operative to the endowment and better- 
ment of man's life." FRANCIS BACON 



ILLUSTRATED BY 

J. F. HORRABIN 



London 
George Allen <SL Unwin Ltd 



FIRST PUBLISHED 1938 

All rights reserved 

Printed in Great Britain 

by 

Unwin Brothers Ltd. 
Woking 



AUTHOR'S CONFESSIONS 

Science for the Citizen is partly written for the large and growing number of 
intelligent adults who realize that the Impact of Science on Society is now the 
focus of genuinely constructive social effort. It is also written for the large 
and growing number of adolescents, who realize that they will be the first 
victims of tie new destructive powers of science misapplied. Since it is the 
first British handbook to Scientific Humanism, it has, inevitably, the glaring 
faults of any new thing. Education segregates the scientific specialist from 
those who study problems of government and social welfare. So, like anyone 
else who in this generation might have attempted a task so ambitious, I 
have had to re-educate myself in the process of writing it. 

Natural science is an essential part of the education of a citizen, because 
scientific discoveries affect the everyday lives of everyone. Hence science 
for the citizen must be science as a record of past, and as an inventory for 
future, human achievements. Inevitably it cannot be divorced from history; 
and my first duty is to acknowledge the patience with which my colleagues 
of the history department in the London School of Economics responded to 
my requests for sources of information. Needless to say, I am not competent 
to judge the reliability of the sources, ^vhich I have quoted at length when 
a personal statement of opinion would imply that I have sufficient first-hand 
knowledge to do so. 

In the Victorian age big men of science like Faraday, T. H. Huxley, and 
Tyndall did not think it beneath their dignity to write about simple truths 
with the conviction that they could instruct their audiences. There were 
giants in those days. The new fashion is to select from the periphery of 
mathematicized hypotheses some half-assimilated speculation as a preface 
to homilies and apologetics crude enough to induce a cold sweat in a really 
sophisticated theologian who knows his job. With a few notable exceptions 
such as Simple Science by Andrade and Huxley and two volumes on British 
and American men of science by J. G. Crowther, this is a fair description 
of the state into which the writing of popular science has fallen hi con- 
temporary Britain. The clue to the state of mind which produces these 
weak-kneed and clownish apologetics is contempt for the common man. The 
key to the eloquent literature which the pen of Faraday and Huxley produced 
is their firm faith in the educability of mankind. 

Because I share that faith, I have not asked the reader to take any reasoning 
on trust. Since the reasoning used in science is often conducted with the sort 
of shorthand called algebra or illustrated by the sort of scale diagrams called 
geometry, a casual glance at isolated pages of this book might be discouraging 
to those who have been humiliated by the obstacles which early education 
places in our path. Fortunately the engaging illustrations of my friend and 
collaborator Horrabin are likely to remove the impression that this is like 
anything that the reader has learned at school. Beyond that let me say this. 
Anyone who has mastered the essence of the first half of Mathematics for the 
Million, or has once been through a course of elementary mathematics leading 



io Science for the Citizen 

to a school-leaving certificate, will have nothing to be afraid of. The rest will 
be told and the reader will be reminded of what may have been long since 
forgotten. 

A presbyterian divine once said that a man who plays golf neglects his 
business, neglects his wife, and neglects his God. Many of the elder states- 
men of science hold that if a younger one writes a book which can be read 
painlessly, he neglects his students, neglects his laboratory, and neglects his 
golf. In self-defence let me therefore explain how this one came to be 
written. 

Sir William Beveridge, Director of the London School of Economics 
during my tenure of the chair of Social Biology in that Institution, held the 
enlightened view that some of the students might benefit from a general 
introduction to natural knowledge. Needless to say, this suggestion was not 
a popular one among those who think that economics is a real science. Still, 
I gave about a hundred lectures, which entailed an unconscionable amount 
of work, considering the small number about twenty all told of brands 
to be saved from the burning. 

I am rather lazy. That is to say, I prefer real work to writing, and only 
write to relieve the tedium of a sickbed, a train journey, or an ocean liner 
load of hearties, empire builders, and wives who are not understood. So I 
usually lecture with a few notes, if any at all. The Southern Railway was 
responsible for seeing that those I made took shape in what is as I hope 
a book which can be read painlessly. How this happened is relevant to a theme 
to which I shall often return. 

Life in South Africa had completely unfitted my children for continuous 
incarceration in the sooty squalid depression of the English metropolis. What 
is mere degradation for a healthy adolescent is plain suicide for a genuine 
naturalist. I believe that biology cannot flourish in an over-urbanized society 
and I also believe that the only beliefs worth burdening oneself with are 
beliefs relevant to the conduct of life. I therefore bought a cottage in Devon 
for a family of four who had grown up barefoot and daylight-conscious. 
With my wife, Enid Charles author of the highly publicized statistics of the 
impending population crisis I moved into a flatlet within two minutes of 
our joint place of work while waiting for a remission of an indeterminate 
sentence. On Friday nights we took an express to rejoin the fast fattening 
four in surroundings which should be the birthright of every British boy or 
girl. Instead of twelve hours occupied each week in exhausting and profitless 
strap-hanging, I now had six hours* carefree time for writing under the most 
propitious circumstances. If the Southern Railway had not provided third- 
class Pullman cars to Exeter, this book would not have been written. 

When I had served my sentence, a typescript of about 2,500 pages was 
just finished. I had no more train journeys to fill up and my living to earn. 
Neither the optimism of the present Master of University College nor the 
amenities of the Southern Railway could now help me to plough through it 
again, to check the calculations or references, and to see it through the press. 
This was done for me. Though I have no resources of generalized "matiness" 
and a sheer genius for making enemies, I have the astonishing good luck to 
get myself liked by young men of promise and with more social gifts than 



Author's Confessions n 

I possess. Three of them torpedoed the typescript into notoriety. Richard 
Palmer, Lecturer in Education at Liverpool University, Dr. J. C. P. Miller, 
Lecturer in Applied Mathematics in the same seat of learning, and Dr. H. D. 
Dickenson, Lecturer in Economic History at Leeds University, appointed 
themselves a sort of intellectual censorship. With heroic disregard for the 
author's bill for excess corrections in proof, they fought every inch of the 
way, when they disagreed with me. Many celebratory paragraphs inserted in 
galley to the despair of the compositors signalize a Pyrrhic victory sometimes 
for me, more often for the Brain Trust. Other friends, including Dr. 
Bacharach, Dr. Clow, and Mr. D. Ridge, made useful comments on separate 
chapters. To all these I am heartily grateful. I will not insert all their 
names, because it might signify their approval of the many passages which 
give the book its entertainment value. Of Horrabin's contribution no word 
is necessary. It speaks for itself. 

Having made it quite clear that this book was produced as the result of 
discharging, and not at the cost of neglecting, my professional work, I may 
as well add that I do not care twopence either way. The surprising state of 
preservation which the Evening Standard describes as my boyish appearance 
is due to the fact that I systematically refuse invitations to dinner parties at 
which people overeat and underthink in costumes designed to inhibit exces- 
sive cerebration. As a boy I earned my pocket money from my father at the 
rate of one halfpenny for thirteen verses of Holy Writ. By the time I was 
eight I could recite 1 Cor. 13 faultlessly. I do not regret it. It taught me that 
when you become a man you put away childish things. Consequently I do 
not play golf. In the time which most men of my profession divide between 
golf and dinner parties, I could write, and have written, several novels, 
which I have had the good taste to burn. 

This nonsense that the scientific worker has no time to be a socially respon- 
sible adult, exercising his social responsibilities as a citizen, is due to be 
debunked. No one lifts an eyebrow if he embraces the ready-made reading 
public of the Gifford Lectures, when he has passed the age of sixty. He is 
then at liberty to demonstrate that the deity is a B star Wrangler with a 
mathematician's contempt for the bread and butter problems of the plain 
man and a mathematician's indifference to the imminent possibility of the 
next war. No one is alarmed except the professional theologian, who generally 
knows his own job rather better. 



LANCELOT HOGBEN 



ABERDEEN 
December, 



CONTENTS 



PART I 

The Conquest of Time Reckoning and 
Space Measurement 

CHAPTER PAGk 

I. POLE STAR AND PYRAMID 

The Coming of the Calendar 1 7 

n. POMPEY'S PILLAR 

The Science of Seafaring 73 

III. SPECTACLES AND SATELLITES 

The Trail of the Telescope 1 3 1 

IV. CHILDREN OF THE SUN 

The Decline of Mere Logic 1 79 

V. WHEEL,, WEIGHT, AND WATCHSPRING 

The Laws of Motion 228 

vi. THE SAILOR'S WORLD VIEW 

The Wave Metaphor of Modern 

Science 314 

PART II 

The Conquest of Substitutes 

VII. THE THIRD STATE OF MATTER 

The Freeborn Miner 357 

VIII. THE REBIRTH OF MATERIALISM 

The Exhaustion of the Neolithic 

Economy 402 

IX. THE ATOMS OF DEMOCRITUS 

Intimations of the Age of Plenty 448 

X. THE LAST RESTING PLACE OF SPIRITS 

A Planned Economy of Carbon Com- 
pounds 496 



14 Science for the Citizen 

PART III 

The Conquest of Power 

CHAPTER PAGE 

XI. A CENTURY OF INVENTIONS 

The Decline of Wind and Water L 547 

XII. THE DARK SATANIC MILLS 

The Superfluity of Mere Toil 582 

XIII. THE SOUTH POINTING INSTRUMENT AND THE 

LIGHTNING CONDUCTOR 630 

XIV. COSTING THE CURRENT 676 
XV. THE WAVES THAT RULE BRITANNIA 736 

PART IV 

The Conquest of Hunger and Disease 

XVI. WHEN LIFE IS CHEAP 775 

XVII. REPLENISHING THE EARTH * 809 

XVIII. THE MICROBE HUNTERS 862 

XIX. THE LAWS OF INCREASING RETURNS 892 

XX. THE ASCENT OF MAN 920 

XXI. A PLANNED ECOLOGY OF HUMAN LIFE 971 

PART V 

The Conquest of Behaviour 

XXII. ANIMAL MAGNETISM 

The Telegraphy of the Body 1013 

XXIII. NATURE AND NURTURE 

Superstitions of Our Own Times 1048 

EPILOGUE. THE NEW SOCIAL CONTRACT 1077 

Answers to Examples 1091 

Index 1099 



PART I 

The Conquest of Time Reckoning 
and Space Measurement 



"THE progress of Greek civilization was dependent essentially 
on the change of slave-labour into free, a transformation not 
supposable without the employment of natural forces, applied 
to labour-saving machines. It is evident that, with the inven- 
tion of a machine which will convert a given natural force 
(e.g. a falling weight of water) into an industrial force, per- 
forming the labour of twenty men, the inventor could grow 
rich and twenty slaves be set free; moreover, that the natural 
effect of the introduction of machines is an augmentation of 
the productive class, whence a greater number of inventors 
and increased production. But, in a slave-state, the appli- 
cation of natural forces and the substitution of machine labour 
for servile, is mainly impossible, for as, in such a state, the 
profits of the capitalist rest upon his slaves, he sees that the 
introduction of machines must imperil his resources, and 
when, as in Greece, the capitalists belong to the ruling 
class, the Government and people will combine to perpetuate 
the existing system, i.e. slavery the Government with the 
seemingly-wise purpose of assuring subsistence to the 
labourers. Only the freeman, not the slave, has a disposition 
and interest to improve implements or to invent them; 
accordingly, in the devising of a complicated machine, the 
workmen employed upon it are generally co-inventors. The 
eccentric and the governor, most important parts of the 
steam-engine, were devised by labourers. The improvement 
of established industrial methods by slaves, themselves 
industrial machines, is out of the question." 

LIEBIG 



CHAPTER I 

POLE STAR AND PYRAMID 

The Coming of the Calendar 

A MUCH abused writer of the nineteenth century said: up to the present 
philosophers have only interpreted the world, it is also necessary to change it. 
No statement more fittingly distinguishes the standpoint of humanistic 
philosophy from the scientific outlook. Science is organized workmanship. 
Its history is co-extensive with that of civilized living. It emerges so soon as 
the secret lore of the craftsman overflows the dam of oral tradition, demanding 
a permanent record of its own. It expands as the record becomes accessible 
to a widening personnel, gathering into itself and coordinating the fruits 
of new crafts. It languishes when the social incentive to new productive 
accomplishment is lacking, and when its custodians lose the will to share 
it with others. Its history, which is the history of the constructive achieve- 
ments of mankind, is also the history of the democratization of positive 
knowledge. This book is written to tell the story of its growth as a record of 
human achievement, a story of the satisfaction of the common needs of 
mankind, disclosing as it unfolds new horizons of human wellbeing which 
lie before us, if we plan our new resources intelligently. 

Whether we choose to call it pure or applied, the story of science is not 
something apart from the common life of mankind. What we call pure science 
only thrives when the contemporary social structure is capable of making 
full use of its teaching, furnishing it with new problems for solution and 
equipping it with new instruments for solving them. Without printing there 
would have been little demand for spectacles; without spectacles neither 
telescope nor microscope; without these the finite velocity of light, the 
annual parallax of the stars, and the micro-organisms of fermentation 
processes and disease would never have been known to science. Without 
the pendulum clock and the projectile there would have been no dynamics 
nor theory of sound. Without the dynamics of the pendulum and projectile, 
no Principia. Without deep-shaft mining in the sixteenth century, when 
abundant slave labour was no longer to hand, there would have been no 
social urge to study air pressure, ventilation, and explosion. Balloons would 
not have been invented, chemistry would have barely surpassed the level 
reached in the third millennium B.C., and the conditions for discovering the 
electric current would have been lacking. 

For this reason the chapters which follow will not adopt the customary 
division of science into separate disciplines, such as chemistry or biology. 
The topics dealt with will be grouped under six main themes; the story of 
man's conquest of time reckoning and earth measurement, of material 
substitutes, of new power resources, of disease, of hunger, and of behaviour. 
When the language of mathematics is used, no advanced knowledge will 
be assumed, and it will present no difficulties if the reader is prepared to do 



i8 Science for the Citizen 

a little work on the examples given. If difficulties arise the reader should 
not be too easily discouraged., or give up hope. If one chapter or page is 
difficult to follow, as likely as not the next will be especially easy. The most 
difficult ones come at the beginning. 

If the execution of the task is novel, there is no originality in the con- 
ception. The reader who is tempted to think so should reflect on the words 
with which the great German chemist Liebig addressed the Royal Academy 
of Sciences at Munich in 1866. Speaking of the Development of Ideas in 
Physical Science, Liebig said: 

The history of physical science teaches us that our knowledge of things 
and of natural phenomena has, for its starting point, the material and intel- 
lectual wants of man and is conditioned by both. . . . Man is not born 
acquainted with sensible objects and their properties and effects; these notions 
must be gained by experience. . . . All these conceptions have sprung or have 
been derived from sensible marks. . . . Since natural phenomena are inter- 
connected like knots in a net, the investigation of particular phenomena evinces 
that they have in common certain conditions, which as remarked are active 
things. . . . Having the facts it is our subsequent business to establish 
their connexion. The facts themselves are obtained through sensual perceptions; 
when these are imperfect, so will be the knowledge reared on them. We can 
have no general theoretical propositions except by means of induction, and 
inductions can be framed only through sensual perceptions. . . . Manifestly 
therefore the truth of explanations does not depend on the principles of logic 
alone. . . . The first explanations can, manifestly, be neither definite nor 
limited, and they must change just in proportion as the facts are more dis- 
tinctly ascertained and as the unknown ones belonging to the conception are 
discovered and incorporated in it. The earlier explanations are therefore only 
relatively false and the latter only therein truer that the contents of the con- 
ceptions of things are more comprehensive, determinate and distinct. . . . 
The conception of time which belongs to the composite notion of velocity was 
first developed fifteen hundred years after Aristotle. For short intervals the 
Greeks had not clocks or time measures. . . Charlemagne's endeavours by 
the establishment of schools to elevate the intelligence of the rude and ignorant 
priesthood of the age could have no result, the soil on which culture thrives 
being not yet prepared. The development of culture, i.e. the extending of man's 
spiritual domain, depends on the growth of the inventions which condition 
the progress of civilization, for through these new facts are obtained. . . . Only 
the free man, not the slave, has a disposition and interest to improve implements 
or to invent them; accordingly^ in the devising of a complicated machine, the 
workmen employed upon it are generally co-inventors. . . . Greek civiliza- 
tion travelled through the Roman Empire and the Arabians into every European 
country. . . . The members of the newly originated intellectual class were at 
first occupied in gaining possession of the treasures of ancient learning. . . . 
The position and employment of the learned of those times concurred in with- 
drawing them from contact with the productive classes. Accordingly the litera- 
ture of that age gives no indication of the degree of the popular civilization and 
culture; for the knowledge circulating through the masses and absorbed into 
their thinking, a knowledge originating in their improved acquaintance with 
physical laws, was not yet stored up in books and was wholly foreign to the 
learned. . . . With the extinction of the slavery of the ancient world and the 
union of all the conditions for the evolution of the human mind, progress 



Pole Star and Pyramid 19 



of civilization and culture is thenceforth assured, indestructible, imperishable. 
Most of the facts from which the investigator elaborated empirical ideas he 
had long since received from the metallurgists, the engineers, the apothecaries, 
and had resolved their inventions into conceptions which the producing classes 
received back . . . the craftsman, technician, agriculturalist, physician, as in 
Greece, ask counsel of the theorist. . . . The history of nations informs us of 
the fruitless efforts of political and theological powers to perpetuate slavery, 
corporeal and intellectual. Future history will describe the victories of freedom 
which men achieved through investigation of the ground of things and of truth, 
victories won with bloodless weapons and in a struggle wherein morals and 
religion participated only as feeble allies. . . . 



THE BEGINNING OF SCIENCE 

We start with the conquest of time and distance. That is to say, the kind 
of knowledge we need to keep track of the seasons and to find our where- 
abouts in the world we inhabit. One depends upon the other. Making a 
calendar and navigating a ship depend on the same kind of knowledge, and 
we shall not be able to keep the two issues apart. Much of the mystery which 
enshrouds contemporary discussion of Relativity will present no difficulty if 
the use of the ship's chronometer is grasped firmly at the outset. All measure- 
ments of time depend on making measurements in space, and localization in 
space depends on measurements of time. 

We used to think of man as a tool-bearing animal, and to divide the pre- 
literate stage of his existence into an old stone age and a new stone age. We 
now know that the social achievements of mankind before the beginning of 
the written record include far more important things than the perfection of 
axes and arrowheads. Three discoveries into which he blundered many 
millennia before the dawn of civilization in Egypt, Sumeria, or Turkestan, are 
specially significant. With the aid of the dogs which followed him and prowled 
about his camp fires, he began to herd instead of to hunt. He learned to 
scatter millet and barley, to store grain to consume when there were no 
fruits to gather. He collected gold nuggets and bits of meteoric iron, and, it 
may be, noticed the formation of copper (see p. 360) from the green pigment 
that he used for adornment, when it was heated in the embers. The sheep 
is an animal with seasonal fertility, and cereal crops are largely annual. In 
domesticating the sheep and learning to sow cereals, man therefore made a 
fateful step. The recognition of the passage of time now became a primary 
necessity of social life. In learning to record the passage of time man learned 
to measure things. He learned to keep account of past events. He made 
structures on a much vaster scale than any which he employed for purely 
domestic use. The arts of writing, architecture, numbering, and in particular 
geometry, which was the offspring of star lore and shadow reckoning, were 
all by-products of man's first organized achievement, the construction of 
the calendar. Shakespeare anticipated Sir Norman Lockyer when he wrote : 
"Our forefathers had no other books but the score and the tally."* 

Science began when man started to plan ahead for the seasons, because 

* Henry VI, Act iv score in this context is from the Anglo-Saxon scora meaning 
notches made on the tally stick. 



2O Science for the Citizen 

planning ahead for the seasons demanded an organized body of continuous 
observations and a permanent record of their recurrence. In an age of wireless 
transmission, of mechanical clocks and cheap almanacs, we take time for 
granted. Before there were any clocks or simpler devices like the hour-glass 
or the clepsydra for recording the passage of time, mankind had to depend 
on the direction of the heavenly bodies, sun by day and the stars by night. 
Already in the hunting and food gathering stage the human race had 
probably learned to associate changes in vegetation, the mating habits of 
animals, and the recurrence of drought or floods, with the rising and setting 




FIG. 1 

The earliest geometrical problems arose from the need for a calendar to regulate the 
seasonal pursuits of settled agriculture. The recurrence of the seasons was recognized 
by erecting monuments in line with the rising s setting, or transit of celestial bodies. 
This photograph, taken at Stonehenge, shows how the position of a stone marked 
the day of the summer solstice when the sun rises farthest north along the eastern 
boundary of the horizon. 

of bright stars and star clusters immediately before sunrise or in evening 
twilight. When the great agrarian revolution reached its climax in the dawn 
of city life, a technique of timekeeping emerged as its pivotal achievement. 
What chiefly remain to record the beginnings of an orderly routine of 
settled life in cities are the vast structures which bear eloquent witness to 
the primary social function of the priesthood as custodians of the calendar. 
The temple, with its corridor and portal placed to greet the transit of its 
guardian star or to trap a thin shaft of light from the rising or setting sun 
of the quarter-days; the obelisk or shadow clock; the Pyramids facing 
equinoctial sunrise or sunset, the pole and the southings of the bright stars 
in the zodiac; the great stone ^circle of Stonehenge with its sight-line 
pointing to the rising sun of the summer solstice all these are first and 
foremost almanacs in architecture. Nascent science and ceremonial religion 



Pole Star and Pyramid 



21 



had a common focus of social necessity in the observatory-temple of the 
astronomer-priest. That we still divide the circle into 360 degrees, that we 

Zenith 





FIG. 2. THE ASTRONOMICAL ORIENTATION OF THE GREAT PYRAMID 

The Pyramid of Cheops and that of Sneferu are constructed on a common geometrical 
plan. The perimeter of the four sides, which face exactly the north, south, east, and 
west, has the same ratio to the height as the ratio of the circumference to the radius 
of a circle, i.e. 2 x 3j, or 277. According to Flinders Petrie: "The squareness and level 
of the base is brilliantly true, the average error being less than a ten-thousandth of the 
side in equality, squareness, and level." At its transit across the meridian, the rays of 
Sirius, the dog star, whose heliacal rising announced the beginning of the Egyptian 
year and the flooding of the sacred river which brought prosperity to the cultivators, 
were at right angles to the south face of the Great Pyramid, and shone straight down 
the ventilating shaft into the royal chamber, illuminating the head of the dead 
Pharaoh. The main opening, and a second shaft leading to the lower chamber, con- 
veyed the light of the Pole Star, which was then the star a in the constellation Draco ., 
at its lower transit, three degrees below the true celestial pole. 

reckon fractions of a degree in minutes and seconds, remind us that men 
learned to measure angles before they had settled standards of length or 



22 Science for the Citizen 

area. Angular measurement was the necessary foundation of timekeeping. 
The social necessity of recording the passage of time forced mankind to map 
out the heavens. How to map the earth came later as an unforeseen result. 

It is a common belief that mathematics is the hallmark of science, and 
some people are apt to imagine that the introduction of a little mathematics 
into subjects like economics entitles them to rank as genuine science. The 
truth is that science rests on the painstaking recognition of uniformities in 
nature. In no branch of science is this more evident than in astronomy, the 
oldest of the sciences, and the parent of the mathematical arts. Between the 
beginnings of city life and the time when human beings first began to sow 
corn or to herd sheep, ten or twenty thousand years perhaps more may 
have been occupied in scanning the night skies and watching the sun's 
shadow throughout the seasons. Mankind was learning the uniformities 
which signalize the passage of the seasons, becoming aware of an external 
order, grasping slowly that it could only be commanded by being obeyed, 
and not as yet realizing that it could not be bribed. There is no hard and 
sharp line between the beginnings of science and what we now call magic. 
Professor Elliot Smith rightly says that magic is the discarded science of 
yesterday. The first priests were also the first scientists and the first civil 
servants. As custodians of the calendar, they created an organized body of 
reliable knowledge from the common experience of herdsman and cultivator. 

To understand how a science of astronomy is possible, we have to acquaint 
ourselves with uniformities of nature, once familiar features of the everyday 
life of mankind. They are no longer part of the everyday life of people who 
live in large cities. So, many readers of this book will need to be told what 
they are. Looking at them retrospectively we can arrange them under four 
headings. 

THE DIURNAL EVENTS 

First we have to reckon with the diurnal events. At daybreak and night- 
fall the shepherd, as he stands at the door of his hut, sees the sun rising in 
different positions at different times of the year, but always towards one 
side of the horizon. He watches it, as it sets in different positions at different 
times of the year, but always towards the opposite side of the horizon. So he 
learns to distinguish an eastern horizon of sunrise and a western horizon 
of sunset. In the region north of the tropics, where the neolithic agrarian 
economy began, the sun travels over the heavens obliquely, so that the 
noon shadow is always on one side of the line joining the eastern and western 
horizons (Fig. 3). The sun's shadow shortens as day wears on till the sun 
itself is highest in the heavens, and then lengthens again as it points more 
and more towards the place of the rising sun. The noon or shortest shadow 
divided the working day of the cultivator into morning and afternoon. 
Fisher folk would be familiar with other time signals besides the daily changes 
of the sun and stars. They would see how high and low water at the tide 
marks would happen twice in a day and a night. Before there was any settled 
husbandry, hunting and food-gathering tribes had learned to recognize 
familiar star clusters, like the Dipper or Great Bear and Cassiopeia, when 



Pole Star and Pyramid 23 

night fell; to know how they change their position like the sun as night goes 
on, and to notice how one star, the Pole Star, is always seen above the same 
point on the horizon, in the same place at sunset and at sunrise. 

When men began to stick up poles or stone pillars to mark off the day by 
the direction and length of the shadow, they would notice that the noon 




TMootv Sun af tine Eouinox 



FIG. 3 

The noon or shortest shadow of the Obelisk or shadow clock points due north towards 
the celestial pole. At the equinoxes (March 21st and September 23rd) the sun rises 
due east and sets due west, and the observer is at the centre of its semicircular 
track, called the equinoctial^ or celestial equator. The angle A which the sun makes 
with the horizon is called its altitude. The angle Z which it makes with the vertical 
is its zenith distance. The altitude of the Pole Star is very nearly constant, and on the 
equinoxes the z.d. of the noon sun is practically equal to the altitude of the Pole 
Star. Hence the plane of the equinoctial is at right angles to the axis which passes 
through the observer and the celestial pole. The stars and moon pass over the horizon 
in circular arcs parallel to that of the sun's transit, rising on the eastern side and 
setting on the western side of the meridian, or ar*c, which passes through the north 
and south points of the horizon, the pole, and the zenith directly overhead. 

shadow always points to the same spot on the horizon, and that the Pole 
Star remains throughout the night exactly above it (Fig. 3). As the night 
passed they would see the other stars revolve about the Pole Star in an anti- 
clockwise direction, from east to west above it, like the sun. They would 
long since have known that star clusters nearest the pole, the "circumpolar" 
stars, never sink below the horizon, trailing from west to east below the pole, 



Science for the Citizen 




Pole Star and Pyramid 



Moon onjan. 1st 

Easterly half seen*. 
Highest above 

liorvzori at sunrise . -c 




. 15 A 



Star 

'before, .sninrise on 
Jan. 15di Jias not , 
yet set at sunrise 
on Jar,. 1st 



lialf seen , v/Jien 
JIOTLZOTL. High&st aiove 
it about sunset. 



FIG. 5. THE CHANGING APPEARANCE OF THE HEAVENS 

In twenty-four hours the whole dome of heaven, including the moon, sun, and fixed 
stars, rotates about an axis which joins the observer to the celestial pole, whose position 
is approximately marked by the Pole Star. With the exception of the "circumpolar" 
stars, which are too near the pole to dip under the horizon, the heavenly bodies all 
appear to rise upwards from the eastern boundary and to sink below the western 
boundary of the horizon plane. In this motion, called the apparent diurnal motion 
of the celestial sphere, the fixed stars retain the same position relative to one another, 
so that at any place the time between the risings or settings of any two stars and the 
direction in which any star is seen rising or setting are always the same. Relative to 
the rising of any fixed star, the moon and the sun each rise a little later on successive 
days. They thus seem to be slipping backwards below the eastern margin of the 
horizon plane. The sun takes 365i days to retreat eastwards till it is again in the same 
position relative to a fixed star, i.e. it slips under the horizon plane eastwards through 
approximately one decree per day. The moon takes 27 J days to do so, but, as the sun 
is slipping back, though more slowly, in the same direction it takes a little longer, 
namely, 29J days (see p. 103) to return to the same position relative to the sun. In the 
figure new moon would occur about January 7th and the next new moon about 
February 5th. At last quarter the moon is 90 west of the sun, rising about midnight 
and reaching its highest point in the heavens about 6 a.m., when its easterly half is 
visible. At first quarter it sets about midnight, reaching its highest point in the heavens 
(meridian transit) at about p.m., when its westerly face is illumined. 



26 



Science for the Citizen 



and from east to west above it (Fig. 4). As the noon shadow divided the day, 
the signal of midnight would be when a star cluster, rising at sunset on the 
eastern horizon and setting at sunrise on the western horizon, was directly 
above the pole. These clusters or "constellations" received fanciful names 
redolent with the preoccupations of everyday life in an agrarian economy. 




FIG. 6. APPARENT DIURNAL MOTION OF THE STARS 

The stars appear to describe circular arcs parallel to one another about an axis which 
joins the observer to the celestial pole. Those nearest the pole the circumpolar stars, 
like A here seen at lower culmination, never sink below the horizon and may, therefore, 
be seen crossing the meridian below the pole. Other northerly stars, such as B, 
describe large arcs over the horizon and so remain above it more than 12 hours between 
rising and setting north of the east and west points. Stars (e.g. C) lying on the 
great equinoctial circle which cuts the east and west points (i.e. stars which rise due 
east and set due west) remain above the horizon for half the 24 hours of the diurnal 
cycle. Stars (like D) which lie south of the equinoctial rise and set towards the southern 
horizon and are below it more than 12 out of the 24 hours. The majority of stars in 
a northern latitude cross the meridian south of the zenith. Hence sailors speak of the 
transit of a star as its "southing." 



Herdsmen watching the night pass would find it just as easy to recognize 
intervals of night time equivalent to the shadow hours of the day. The much 
despised yokel, who has not upholstered his brain with the urban super- 
stitions of all the ages,, is often adept in using the star clock. A little practice 
while camping out is sufficient to enable you to tell the time by the stars 
with an error scarcely more than quarter of an hour. 

Centuries before city life began, man had begun to fumble for a connected 
account of the regularities forced on his attention. He already knew that 



Pole Star and Pyramid 27 

sun, moon, and stars partake of the same daily and nightly motion about 
one central point in the heavens. As they stand, the facts with which primitive 
man was familiar in his everyday life are capable of being looked at in two 
ways. When we are passing another train, we cannot at once tell whether 
we are at rest and the other one is in motion, or vice versa. So we cannot tell 
whether we are going east to meet the sun and rising stars, or whether they 
are moving west to meet us. In the train we can settle the issue by looking 
out of the opposite window. We put ourselves in the position of the man on 
the platform. Primitive man had no knowledge of what the two trains would 
look like from the platform. Having no other court of appeal, he inclined to 
his first impression that the sun and stars were rushing past him. 

In the priestly lore of the earliest calendar civilizations the picture pieced 
together was something like this. The stars, moon, and sun were all on the 
surface of a great sphere, of which we only see one half at a time. The stars 
become visible when the sun is in the celestial hemisphere below our 
horizon (Fig. 5). The celestial sphere revolves around an axis joining the 
Pole Star (Fig. 6) to some fixed spot on the earth the North Pole, as we call 
it today. It completes its revolution in a day and a night, revolving in an 
anti-clockwise direction from the standpoint of a person looking upwards 
towards the North Pole Star. 

Although the facts are equally explicable on the alternative assumption 
that the stars are fixed and that the earth is revolving about the same axis in 
the opposite direction to the apparent revolution of the celestial sphere, the 
earlier and less sophisticated view embodied a tremendous gain. It involved 
the first step towards a world map. In counting the shadow hours and learn- 
ing to use the star clock, man had begun to use geometry. He had begun to 
find his local bearings in cosmic and terrestrial space. An important step 
towards an art of measurement was made when men began to trace circles on 
the sand or the soft earth around the shadow pole to mark the moment when 
the shadow was shortest. In discovering the constant direction of the noon 
shadow pointing to the pole, they fixed two planes of reference. One is the 
horizontal plane, the north and south points of which divide the observer's 
terrestrial horizon into an easterly and a westerly half. The other was bounded 
by the great semicircle or Meridian of the heavens, with its highest point, 
the zenith, directly overhead (Fig. 3). The axle of the heavenly clock of star 
transit and shadow connected the Pole Star to the earthly pole in the meridian 
plane. Sun, moon, and stars are highest in the heavens where the circles 
they describe on the surface of the celestial sphere cut the meridian, 

THE MONTHLY EVENTS 

Strictly speaking in order of time, the first class of uniformities from which 
the measurement of time proceeded were in all probability the lunar 
phenomena, from which we got the grouping of days into months and weeks 
(quarter months). There are still backward peoples who have not learned 
to reckon in years of equivalent length. The recognition of the month is 
wellnigh universal even among hunting tribes with no settled agriculture. 
Moonlight is a circumstance of enormous importance in the everyday life 
of people who have crude means of artificial illumination. Even today in 



28 Science for the Citizen 

remote parts of the country the time of full moon is chosen for a long night 
journey. 

An interval of roughly thirty days separates one full moon or new moon 
from another. The two half moons, the first "quarter" when waxing and the 
third "quarter" when waning, complete the division of the month into 
quarters, which roughly correspond to our week. Near the sea it is noticed 
that the tides are exceptionally high when the moon is invisible through the 




FIG. 7. THE MONTHLY CYCLE OF THE MOON'S PHASES 

whole night (new moon) and when it is full. At first and third quarter (half 
moons) the high-water mark is exceptionally low. The most important 
thing (Fig. 5) connected with the changing appearance of the moon is that 
as the moon waxes and wanes it rises towards the east a little later every 
day. At first quarter (Fig. 7) it is already high in the heavens at sunset, 
setting about midnight. The full moon rises about sunset, is at its highest 
about midnight, and sets towards sunrise. At the third quarter the waning 
moon does not rise till about midnight, is seen at its highest point ("crosses 
the meridian") about sunrise, and is visible during the morning by daylight.* 
The moon seems to partake of the general motion of the celestial sphere, 

* The relative times of setting and rising of the sun and moon depend partly on 
their declinations, as explained HI Chapter II, and are different at different seasons. 
These remarks, like others on the next page, are only to be taken as a rough outline 
of what happens. 



Pole Star and Pyramid 



Sirius, the brightest star in the sky. Sirius is a winter star, rising at sunset 
about the beginning of January. Early in March it is already setting by 
midnight. After being invisible throughout the night in June, Sirius reappears 

Sun, transits /ugfc 
above famzoti ,June ZLst. 



' ^4^^/^& (Sun, in, tyias rises 




E. (in 

/r \ 

Capnconi) 



(Sltt^ in- Libra, vises due. 



FIG. 9. THE CHANGING HEAVENS 

The successive positions of the sun in the heavens during its annual retreat below the 
eastern horizon in the circle called the ecliptic were mapped out by the ancient priest- 
hoods in milestones corresponding to the twelve months of the year. These mile- 
stones, the zodiacal constellations^ were groups of stars whose rising and setting 
positions roughly corresponded to that of the sun at a particular season. Owing to 
the slow rotation (precession of the equinoxes) of the equinoctial circle about the ecliptic, 
the sun's position among the fixed stars at a particular season is not the same as it 
was in ancient times, here shown. When the sun occupies the position of Aries (i.e. is 
seen in the same direction as Aries would be seen if visible), it sets and rises with the 
latter, which is therefore invisible. A month later, when the sun is in Taurus, Taurus 
rises and sets with the sun and is invisible. Aries is seen rising just before sunrise 
where the sun rose a month earlier. When the sun was in Aries, Taurus would have 
been setting for about an hour after sunset where the sun would sink below the horizon 
a month later. The constellations corresponding to the sun's position during the 
summer months (Taurus and Virgo, Gemini and Leo, Cancer) had northerly risings 
and settings, describing large arcs and therefore remaining long above the horizon in 
the winter night sky. The constellations mapping put the sun's position in the winter 
months (Pisces and Scorpio, Aquarius and Sagittarius, Capricorn) have southerly 
risings and settings, describing short arcs above the horizon and being conspicuous 
during the short summer nights. (See also Figs. 138 and 166.) 



32 Science for the Citizen 

on the eastern horizon a few minutes before sunrise on a day in July. This 
happened at the time when the flooding of the Nile brought assurance 
of food and prosperity to the Middle Kingdom. The advanced state of 
astronomical knowledge in the calendar civilizations of antiquity need not 
surprise us, when we take stock of the astronomical knowledge of living 
peoples whose cultural development is in other respects very primitive. The 
following extracts from Nilsson's* monograph Primitive Time Reckoning 
(pp. 109-143), are instructive : 

Time-indications from the phases of the climate and of Nature are only 
approximate: they themselves, like the concrete phenomena to which they 
refer, are subject to fluctuation. ... In general, primitive man takes no notice 
of these variations : the Banyankole, for instance, are indifferent as to whether 
the year is one or even three weeks longer or shorter, i.e. whether the rainy 
season opens so much earlier or later. The days are not counted exactly, but the 
people are content with the concrete phenomenon. More accurate points of 
reference are, however, especially desirable for an agricultural people, since, 
although the right time for sowing can be discerned from the phenomena and 
general conditions of the climate, yet a more exact determination of time may 
be extremely useful. The possibility of such a determination exists and that 
at a far more primitive stage than that of the agricultural peoples in the 
observation of the stars, and especially in the observation of the so-called 
" apparent " or, more properly, visible risings and settings of the fixed stars, 
the importance of which has already been explained. The observation of the 
morning rising and evening setting is extraordinarily widespread, but other 
positions of the stars, e.g. at a certain distance from the horizon, are also some- 
times observed. The Kiwai Papuans also compute the time of invisibility of a 
star. When a certain star has sunk below the western horizon they wait for some 
nights during which the star is "inside"; then it has "made a leap," and shows 
itself in the east in the morning before sunrise. . . . Stellar science and 
mythology are therefore widespread among the primitive and extremely 
primitive peoples, and attain a considerable development among certain bar- 
baric peoples. Although this must be conceded, some people are apt to think 
that the determination of time from the stars belongs to a much more advanced 
stage : it is frequently regarded as a very learned and very late mode of time- 
reckoning. Modern man is almost entirely without knowledge of the stars; for 
him they ^'?e the ornaments of the night-sky, which at most call forth a vague 
emotion or are objects of a science which is considered to be very difficult 
and highly specialized, and is left to the experts. It is true that the accurate 
determination of the risings and settings of the stars does demand scientific 
work, but not so the observation of the visible risings and settings. Primitive 
man rises and goes to bed with the sun. When he gets up at dawn and steps 
out of his hut, he directs his gaze to the brightening east, and notices the stars 
that are shining just there and are soon to vanish before the light of the sun. 
In the same way he observes at evening before he goes to rest what stars appear 
in the west at dusk and soon afterwards set there. Experience teaches him that 
these stars vary throughout the year, and that this variation keeps pace with 
the phases of Nature, or, more concretely expressed, he learns that the risings 
and settings of certain stars coincide with certain natural phenomena. . . . 
Just as the advance of the day is discerned from the position of the sun, so 
the advance of the year is recognized by the position of certain stars at sunrise 
* Acta Soc. Human. Litt. Lundensis, 1920. 



Pole Star and Pyramid 33 



and sunset. Stars and sun alike are the indicators of the dial of the heavens. 
A determination of this kind, however, is not so accurate as that from heliacal 
risings and settings. Hence the latter pass almost exclusively or at least pre- 
eminently under consideration wherever, as in Greece, a calendar of the natural 
year is based upon the stars: sometimes, however, the upper culmination is 
given. ... In order to determine the time of certain important natural 
phenomena it is therefore sufficient to know and observe a few stars or con- 
stellations with accuracy and certainty. The Pleiades are the most important. 
It has been asked why this particular constellation, consisting as it does of 
comparatively small and unimportant stars, should have played so great a part, 
and the answer is chiefly that its appearance coincides (though this is true of 
other stars also) with important phases of the vegetation. . . . An account of 
the Bushmen shows how extremely primitive peoples may also observe the 
risings of the stars, may connect them with the seasons, and which is indeed 
somewhat rare may even worship them. . . . Canopus and Sirius appear 
in winter, hence the cold is connected with them. . . . The Hottentots con- 
nect the Pleiades with winter. These stars become visible in the middle of 
June, that is, in the first half of the cold season, and are therefore called "Rime- 
stars," since at the time of their becoming visible the nights may be already 
so cold that there is hoar-frost in the early morning. The appearance of the 
Pleiades also gives to the Bushmen of the Auob district the signal for departure 
to the tsama field. ... A tribe of Western Victoria connected certain con- 
stellations with the seasons. . . . The winter stars are Arcturus who is held 
in great respect since he has taught the natives to find the pupae of the wood- 
ants, which are an important article of food in August and September and 
Vega, who has taught them to find the eggs of the mallee-hen, which are also 
an important article of food in October. The natives also know and tell stories 
of many other stars. Another authority states that they can tell from the position 
of Arcturus or Vega above the horizon in August and October respectively 
when it is time to collect these pupae and these eggs. . . . For example, when 
Canopus at dawn is only a very little way above the eastern horizon, it is time 
to collect eggs; when the Pleiades are visible in the east a little before sunrise, 
the time has come to visit friends and neighbouring tribes. The Chukchee form 
out of the stars Altair and Tarared in Aquila a constellation named pchittin, 
which is believed to be a forefather of the tribe who, after death, ascended into 
heaven. Since this constellation begins to appear above the horizon at the time 
of the winter solstice, it is said to usher in the light of the new year and most 
families belonging to the tribes living by the sea bring their sacrifices at its 
first appearing. . . . The South American Indians have much greater know- 
ledge of the stars, and in consequence frequently connect stellar phenomena, 
especially those of the Pleiades, with phases of Nature. In north-west Brazil 
the Indians determine the time of planting from the position of certain con- 
stellations, in particular the Pleiades. If these have disappeared below the 
horizon, the regular heavy rains will begin. The Siusi gave an accurate account 
of the progress of the constellations, by which they calculated the seasons, and in 
explanation drew three diagrams in the sand. No. 1 had three constellations: 
"a Second Crab," which obviously consists of the three bright stars west of 
Leo, "the Crab," composed of the principal stars of Leo, and "the Youths," 
i.e. the Pleiades. When these set, continuous rain falls, the river begins to 
rise, beginning of the rainy season, planting of manioc. No. 2 had two con- 
stellations: "the Fishing Basket," in Orion, and kakudzuta, the northern 
part of Eridanus, in which other tribes see a dancing implement. When these 

B 



34 Science for the Citizen 



set, much rain falls, the water in the river is at its highest. No. 3 was "the 
Great Serpent," i.e. Scorpio. When this sets there is little or no rain, the water 
is at its lowest. The natives of Brazil are acquainted with the course of the con- 
stellations, with their height and the period and time of their appearance in, 
and disappearance from the sky, and according to them divide up their seasons. 
... In Africa also the observation of the stars, and above all the Pleiades, is 
widespread. In view of the dissemination of this knowledge all over the world 
it is making a quite unnecessary exception to state that it came into Africa 
from Egypt. Moreover, this assertion does not correspond with the facts, since 
among the Egyptians Sirius, and not the Pleiades, occupied the chief place. 
. . . The Melanesians of Banks Island and the northern New Hebrides are 
also acquainted with the Pleiades as a sign of the approach of the yam-harvest. 
The inhabitants of New Britain (Bismarck Archipelago) are guided in ascer- 
taining the time of planting by the position of certain stars. The Moanu of the 
Admiralty Islands use the stars as a guide both on land and at sea, and recognize 
the season of the monsoons by them. When the Pleiades (tjasd) appear at 
nightfall on the horizon, this is the signal for the north-west wind to begin. 
But when the Thornback (Scorpio) and the Shark (Altair) emerge as twilight 
begins, this shows that the south-east wind is at hand. When the "Fishers' 
Canoe" (Orion, three fishermen in a canoe) disappears from the horizon at 
evening, the south-east wind sets in strongly: so also when the constellation 
is visible at morning on the horizon. When it comes up at evening, the rainy 
season and the north-west wind are not far off. When "the Bird" (Canis major) 
is in such a position that one wing points to the north but the other is still 
invisible, the time has come in which the turtles lay eggs, and many natives 
then go to the Los-Reys group in order to collect them. The Crown is called 
the "Mosquito-star," since the mosquitoes swarm into the houses when this 
constellation sets. The two largest stars of the circle are called pitui and papal i 
when this constellation becomes visible in the early morning, the time is 
favourable for catching the fish papai. The natives of the Bougainville Straits 
are acquainted with certain stars, especially the Pleiades: the rising of this 
constellation is a sign that the kai-nut is ripe : a ceremony takes place at this 
season. On Treasury Island a grand festival is held towards the end of October, 
in order so far as could be ascertained to celebrate the approaching appear- 
ance of the Pleiades above the eastern horizon after sunset. In Ugi, where of all 
the stars the Pleiades alone have a name, the times for planting and taking up 
yams are determined by this constellation. In Lambutjo the year is reckoned 
according to the position of the Pleiades. . . . When the stars indicate this or 
that event, the primitive mind, as so often happens, is unable to distinguish 
between accompanying phenomena and causal connexion; it follows that the 
stars are regarded as authors of the events accompanying their appearance, when 
these take place without the interference of man. So in ancient Greece the ex- 
pressions (a certain star) "indicates" or "makes" certain weather were not kept 
apart, and the stars were regarded as causes of the atmospheric phenomena. 
A similar process of reasoning is not seldom found among primitive peoples, 
and a few instances have already been given, such as the warning-incantation 
of the Bushmen against Canopus and Sirius, the name given to the Pleiades 
among the Bakongo ("the Caretakers-who-guard-the-rain"), and the belief 
that the rain comes from them, the myth of the Euahlayi tribe that the 
Pleiades let ice fall down on to the earth in winter and cause thunderstorms, in 
other words send the rain, and the belief of the Marshall Islanders that the 
various positions of certain stars cause storms or good winds. 



Pole Star and Pyramid 35 

These extracts illustrate how familiar facts in the lives of food-gathering 
and primitive agrarian communities prepare the way for a fixed calendar 
and the emergence of a temple culture. The separation of a caste entrusted 
with the social responsibility of regulating the seasonal pursuits of a settled 
agrarian economy marks the beginning of written history. Only at this 
point does the need for a permanent record of events and measurements 
emerge. Here, also, we see history repeating itself or if you prefer it 
history at a standstill in backward cultures of the present day. Speaking of 
the time-keeping function of the priesthood in contemporary societies, 
Nilsson says (pp. 347-354): 

As long as the determination of time is adjusted by the phases of Nature 
which immediately become obvious to everyone, anybody can judge of them, 
and should different people judge differently there is no standard by which the 
dispute can be settled, because the natural phases run into one another or are 
at least not sharply defined. The accuracy in determination demanded by 
time-reckoning proper is therefore lacking. Accuracy becomes possible as a 
result of the observation of the risings of stars, and this observation begins even 
at the primitive stage, but it is not a matter that concerns everyone. It requires 
a refined power of observation and a clear knowledge of the stars, so that the 
heavens can be known. This is especially the case with the commonest observa- 
tions, those of the morning rising and evening setting. The observer must be 
able to judge, by the position of the other stars, when the star in question 
may be expected to twinkle for a moment in the twilight before it vanishes. 
The accuracy of the time-determination from the stars depends therefore upon 
the keenness of the observation. In this the individual differences of men soon 
come into play, along with a regular science which introduces the learner to 
the knowledge of the stars and its uses. Thus Stanbridge reports of the natives 
of Victoria that all tribes have traditions about the stars, but certain families 
have the reputation of having the most accurate knowledge; one family of the 
Boorung tribe prides itself upon possessing a wider knowledge of the stars than 
any other. ... By the phases of the stars both occupations and seasons are 
regulated, and thus a standard is furnished by which to judge, and a limit is 
set to the indefiniteness of the phases of Nature. . . . The moon strikes the 
attention of everyone and admits of immediate and unpractised observation; 
at the most there may sometimes be some doubt for a day as to the observa- 
tion of the new moon, but the next day will set all right. But because the months 
are fixed in their position in the natural year through association with the 
seasons, the indefiniteness and fluctuation of the phases of Nature penetrate 
into the months also, and are there even increased, for the reasons stated 
above. Cause for doubt and disagreement is given, the problem of the regula- 
tion of the calendar arises. Hence in the council meetings of the Pawnee and 
Dakota it is often hotly disputed which month it really is. So also the Caffres 
often become confused and do not know what month it is; the rising of the 
Pleiades decides the question. The Basuto in determining the time of sowing 
are not guided by the lunar reckoning, but fall back upon the phases of Nature; 
intelligent chiefs, however, know how to correct the calendar by the summer 
solstice. . . . The differences in intelligence already make themselves felt at an 
early stage, and are still more plainly shown when we come to a genuine regula- 
tion of the calendar. Some of the Bontoc Igorot state that the year has eight, 
others a hundred months, but among the old men who represent the wisdom 
of the people there are some who know and assert that it has thirteen. The 



Science for the Citizen 



further the calendar develops, the less does it become a common possession. 
Among the Indians, for example, there are special persons who keep and 
interpret year-lists illustrated with picture-writings, e.g. the calendrically 
gifted Anko, who even drew up a list of months. It is very significant that even 
where a complete calendar does exist, it will be found that this is not in use 
to its fullest extent among the people. ... It follows that the observation 
of the calendar is a special occupation which is placed in the hands of specially 
experienced and gifted men. Among the Caffres we read of special "astrologers." 
Among the Kenya of Borneo the determination of the time for sowing is so 




on June. 21 



Due 



on Dec. 21. 



FIG. 10 



This figure shows how the sun's apparent track in the heavens varies with the seasons 
The noon shadow is shortest, i.e. the noon sun is highest, on June 2 1st. The noon 
shadow is longest, i.e. the noon sun is at its lowest, on December 21st. At the 
equinoxes, midway between, the sun rises due east and sets due west. Owing to 
distortion in a plane figure, the noon shadow is relatively larger than it should be. 
In reality, of course, the shadow is always shortest and always points due north at 



important that in every village the task is entrusted to a man whose sole occupa- 
tion it is to observe the signs. He need not cultivate rice himself, for he will 
receive his supplies from the other inhabitants of the village. His separate 
position is in part due to the fact that the determination of the season is effected 
by observing the height of the sun, for which special instruments are required. 
The process is a secret, and his advice is always followed. It is only natural that 
this individual should keep secret the traditional lore upon which his position 
depends; and thus the development of the calendar puts a still wider gap 
between the business of the calendar-maker and the common people. Behind the 



Pole Star and Pyramid 



37 



calendar stand in particular the priests. But they are the most intelligent and 
learned men of the tribe, and moreover the calendar is peculiarly their affair 
if the development has proceeded so far that value is attached to the calendar 
for the selection of the proper days for the religious observances. Among the 
priests there is formed a special class whose duty it is to make observations and 
keep the calendar in order. Among the Hawaiians "astronomers (kilo-hoku) 




FIG. 11. FIXING THE MERIDIAN IN ANCIENT TIMLS 

The method of equal altitudes was used to get the direction of the north point exactly. 
The points at which this shadow just touched a circle traced on the sand around the 
shadow clock shortly before and after noon were noted, and the arc was bisected. 



and priests" are mentioned; they handed down their knowledge from father to 
son; but women, kilowahine, are also found among them. Elsewhere the nobles 
appear alongside of the priests; thus in Tahiti it is the nobles who are respon- 
sible for the calendar, in New Zealand the priests. In the latter country there 
is said to have been a regular school, which was visited by priests and chiefs 
of highest rank. Every year the assembly determined the days on which the 
corn must be sown and reaped, and thus its members compared their views 
upon the heavenly bodies. Each course lasted from three to five months. 



38 Science for the Citizen 

Wherever we find a calendar priesthood in existence among contemporary 
peoples in a backward state of culture, or in ancient civilizations as far removed 
as the Megalithic of Stonehenge and the Maya temples of Guatemala, 
knowledge of a second group of phenomena based on the sun's behaviour 
grows side by side with and tends to displace the reckoning of the year by 
the rising and setting of stars. The sun's noon shadow waxes and wanes. 
In using the noon shadow as a time marker, man learned to distinguish four 
days which have a characteristic relation to the seasonal changes of wind, 



Sun 




FIG. 12. FIXING THE DAY OF THE EQUINOCTIAL FESTIVAL 

Some early calendrical monuments suggest that the equinox was fixed by obser- 
vations on the rising or setting sun of the solstices (December 21st and June 21st), 
when the sun rises and sets at its most extreme positions towards the south and north 
respectively. In the figure, A and B are two poles placed in alignment with the setting 
sun of the winter solstice. The distance between A and C in line with the setting sun 
of the summer solstice is the same as the distance between A and B. Midway on 
its journey between the two extremes the sun rises and sets due east and west, and 
the lengths of day and night are equal. Hence these two days (March 21st and Sep- 
tember 23rd) are called the equinoxes. In ancient ritual they were days of great 
importance. The east and west points on the horizon can be obtained by bisecting the 
angle BAG. 

rainfall, and warmth. The length of the day is longest when the noon shadow 
is shortest and the sun travels round the heavens perceptibly nearer the 
northern hah of the horizon on the summer solstice (June 21st in our 
calendar) (Fig. 10). The length of the day is appreciably shortest when the 
noon shadow is longest and the sun rises and sets furthest towards the 
southern limit of the horizon on the winter solstice (December 21st). Midway 
between these two dates are two days on which the hands of the shadow 
dock describe a semicircle. On these two days, the vernal equinox (March 21st) 
and the autumnal equinox (September 23rd), day and night are of equal 
length. The sun rises exactly halfway between the north and south points 
of the horizon on the east side and sets exactly halfway between the north 



Pole Star and Pyramid 



39 



and south points of the horizon on the west side. Due east is the position 
of the rising sun on the equinoxes. Due west is the position of the setting 
sun on the same days. The surface bounded by the track of the equinoctial 
sun from the east and west points of the horizon gave the priestly custodians 
of the calendar a third plane of reference at right angles to the earth's axis 
(see Figs. 3, 6, and 9). It is called the celestial equator or equinoctial. 
One of the earliest problems in., the practical geometry of a calendar 

-June 21 s - -Sun in. Cancer- Cazuxzr 
& sets WI$L th& sun , & is invtsiblz ~kkrc 
i nioht . (jenwu. Tis'uiff in- east oertwe. 

. o ~* . . O 

sunrise, i&o setting- tn w&st after sunsetr. 

3O d^ys Zafer 

rises <^ sobs 
is invisible i 

the, ruglii: . Can-Cer rising 
6^? /ore 




East- 



in. 



FIG. 13. THE SUN'S (APPARENT) ANNUAL RETREAT THROUGH THE 
ZODIACAL CONSTELLATIONS 

(The South Pole of the earth is nearest to you in the figure.) 

priesthood arose in watching for the return of the equinoxes. One way in 
which the priests of antiquity fixed the exact direction of the meridian is 
shown in Fig. 11. With a sufficiently long piece of cord fairly high accuracy 
can be secured. Laying off the east and west points of the horizon to record 
the equinoxes was probably done in a similar way, two poles or stones being 
erected in line with the rising or setting sun of the summer solstice, and a 
third equidistant from one of them in line with it and the rising or setting 
sun of the winter solstice (Fig. 12). The sun of the equinoxes would rise 
and set along the line bisecting the angle between the sun's positions on the 



40 Science for the Citizen 

solstices. The Egyptians already recognized that this could also be done by 
making a line at right angles to the meridian. The division of the daily shadow 
path into hour angles was a later device probably of Babylonian origin, and 
betrays the early connexion between the art of space measurement and the 
social necessity of recording the passage of time. The division of the equinoc- 
tial half-circle into twelve divisions is not surprising. Of all integral sub- 
multiples of 360 the angle 15 is the smallest whole number which we can 




FIG. 14 

Noon on the solstices at lat. 45 N. On June 21st the sun is highest in the heavens, 
i.e. nearest the zenith. Its zenith distance (Z,) is least. On December 21st the sun is 
lowest in the heavens furthest from the zenith. Its zenith distance (Z 2 ) is greatest. 
The angle E is the angle which the sun's apparent annual track (ecliptic) makes with 
the equator (equinoctial) of the celestial sphere. On March 2 1st and September 23rd 
day and night are of equal length, and the sun appears to lie on the celestial equator 
at noon. Note its zenith distance is then practically equivalent to the observer's latitude. 
(Compare Figs. 15 and 25.) 

easily make by elementary methods of construction. By knotting cords at 
equal lengths we can peg out an equilateral triangle. Successive bisection of 
the angles of the equilateral triangle then gives 30 and 15, 

The phenomena of the rising and setting of stars show that the sun 
changes its position relative to the fixed stars, as if retreating eastwards 
through a complete circle in the celestial sphere. To account for the changing 
height of the noonday sun and the duration of the days and nights throughout 
the year, a second conception took shape. The sun appears to slip back 
through a track, the ecliptic, which is placed obliquely with reference to 
the polar axis (Figs. 9, 14). By about three thousand B.C. we have ample 



Pole Star and Pyramid 41 

evidence that the priests of Egypt had constructed simple instruments for 
measuring the angular direction of the stars, and were accustomed to watch 
for the moment when a star crosses the meridian, i.e. the great semicircle 
which cuts the north horizon, the Pole Star, the zenith vertically above the 
observer, and the south horizon. By noting the direction of the sun from 
the south horizon when it crosses the meridian at noon, they were able to 
identify the sun's annual track through a belt of twelve star clusters, called 
the Zodiac, corresponding to the twelve 30-day months of the Babylonian 



Pole Star- 



. 
Spring "Earth triN&v / 

- 




\ Sr?- _>_!_ 

\ %-tor ^ 



FIG. 15 

If the stars arc immensely distant the sun's apparent track through the ecliptic is 
equally explicable if we assume that the earth really moves annually round the sun 
with its North Pole always pointing towards the celestial pole and the plane of its 
equator always parallel to that of the celestial equator in an orbit which is oblique to 
the latter. (Compare Fig. 14.) 

year (Fig. 13). The star clusters of the Zodiac are not systems of bodies 
with any known relation among themselves. They are simply signposts of 
the seasons. The times of rising and setting of a zodiacal constellation and 
its height above the southern horizon when it crosses the meridian correspond 
fairly closely with the times of sunrise and of sunset and with the height of 
the noon sun six months earlier or later (see Fig. 9). The names of the Zodiac 
star clusters are : Aries, Taurus, Gemini, Cancer, Leo, Virgo, Libra, Scorpio, 
Sagittarius, Capricornus, Aquarius, Pisces. 

That two of these names are familiar to all of us draws attention to the 
far-reaching importance of the hypothesis which gradually developed from 

B* 



42 Science for the Citizen 

this foundation. It is true that all the facts are equally intelligible on another 
view, if we bear in mind what we now know about the immense distance 
of the fixed stars. The sun's apparent track through the ecliptic is also 
explicable if we assume that our train is moving and the sun's engine is at 
rest. All that we can see is compatible with the more sophisticated, and for 
the present purpose less straightforward, hypothesis that the earth pursues 
a slanting annual track around the sun with its polar axis always at the same 
angle to the ecliptic plane (Fig. 15). On either view we have made a very 
big advance in our knowledge of the earth through widening our knowledge 
of the heavens. We shall see this better when we have taken into account 
another class of events which clarified the recognition that our earth itself 
is a spherical body. 

With regard to the formative influence which the calendar exerted in the 
early stages of civilization, the important point to grasp is that the determin- 
ation of the year as the interval between successive summer or winter 
solstices or alternate equinoxes demands attention to measurement. Thus 
Nilsson, speaking of contemporary communities, says (pp. 311-318): 

An observation of the annual course of the sun, therefore, unlike that of 
the stars which everywhere, no matter where, can be performed imme- 
diately demands a fixed place and special aids to determination. It follows 
that the observation of the solstices and equinoxes belongs to a much higher 
stage of civilization than does that of the stars. ... It is used by the Eskimos, 
who have a very highly developed sense of place, and know how to make good 
maps. Moreover, where the sun in winter stands very low on the horizon, and 
for a time altogether disappears beneath it, the conditions are very favourable 
for the observation of its return. Older authors say that by the rays of the sun 
on the rocks the Eskimos can tell with tolerable accuracy when it is the shortest 
day; more recently we have been told of the Arnmasalik that they can calculate 
beforehand the time of the shortest day and that accurately to the day not 
only from the solstitial point, but also from the position of Altair in the morning 
twilight. They begin their spring when the sun rises at the same spot as Altair. 
. . . The Incas erected artificial marks. There were in Cuzco sixteen towers, 
eight to the west and eight to the east, arranged in groups of four. The two 
middle ones were smaller than the others, and the distance between the towers 
was eight, ten or twenty feet. The space between the little towers through 
which the sun passed at sunrise and sunset was the point of the solstices. In 
order to verify this the Inca chose a favourable spot from which he observed 
carefully whether the sun rose and set between the little towers to east and 
west. For the observation of the equinoxes richly ornamented pillars were set 
up in the open space before the temple of the sun. When the time approached, 
the shadow of the pillars was carefully observed. The open space was circular, 
and a line was drawn through its centre from east to west. Long experience had 
taught them where to look for the equinoctial point, and by the distance of the 
shadow from this point they judged of the approach of the equinox. When 
from sunrise to sunset the shadow was to be seen on both sides of the pillar and 
not at all to the south of it, they took that day as the day of the equinox. This 
last account is for Quito, which lies just under the equator. At the spring 
equinox the maize was reaped and a feast was celebrated, at the autumn equinox 
the people celebrated one of their principal feasts. The months were calculated 
from the winter solstice. . . . One would suspect that this Melanesian science, 



Pole Star and Pyramid 43 

like the knowledge of the stars, is borrowed from the Polynesians : for the latter 
understood the annual course of the sun. In Tahiti the place of the sunrise was 
called tataheita, that of the sunset topa-t-era. The annual movement of the 
sun from the south towards the north was recognized, and so was the fact that 
all these points of the daily approach to the zenith lay in a line. This meridian 
was called t'era-hwattea, the northern point of it tu-errau^ and the opposite 
point above the horizon, or the south, toa. According to other sources the 
December solstice was called rua-maoro or rua-roa, the June solstice rua-poto. 
The Hawaiians called the northern limit of the sun in the ecliptic "the black, 
shining road of Kane," and the southern limit "the black, shining road of 
Kanaloa." The equator was named "the bright road of the spider" or "the 
road to the navel of Wakea," equivalent to "the centre of the world." How 
the Polynesians came to recognize the tropics and the equator is unfortunately 
unknown, but certainly they did it like other peoples by observing the solstices 
and equinoxes at certain landmarks. . . . Agricultural peoples in particular 
have developed various methods of this kind. The rice-cultivating peoples of 
the East Indies use various methods in order to determine the important 
time of sowing. Of the observation of the stars we have already spoken. Among 
the Kayen of Sarawak an old priest determines the official time of sowing from 
the position of the sun by erecting at the side of the house two oblong stones, 
one larger and one smaller, and then observing the moment when the sun, in 
the lengthening of the line of connexion between these two stones, sets behind 
the opposite hill. The sowing-day is the only one determined by astronomical 
methods. In other respects the time-reckoning is a more or less arbitrary one, 
and is dependent on the agriculture. Of the hollows in a block of stone at Batu 
Sala, in the river-bed of the upper Mahakam, it is said that they originated in 
the fact that the priestesses of the neighbouring tribes used formerly to sit 
on the stone every year in order to observe when the sun would set behind a 
certain peak of the opposite mountain. This date then decided the time for the 
beginning of the sowing. . . . The Kenyan observe the position of the sun. 
Their instrument is a straight cylindrical pole of hardwood, fixed vertically 
in the ground and carefully adjusted with the aid of plumb-lines; the possi- 
bility of its sinking deeper into the earth is prevented. The pole is a little 
longer than the outstretched arms of its maker and stands on a cleared space 
by the house, surrounded by a strong fence. The observer has further a flat 
stick on which lengths measured from his body are marked off by notches. 
The other side has a larger number of notches, of which one marks the greatest 
length of the midday shadow, the next one its length three days after it has 
begun to shorten, and so on. The shadow is measured every midday. As it 
grows shorter after reaching its maximal length the man observes it with special 
care, and announces to the village that the time for preparing the land is near 
at hand. In Bali and Java the seasons are determined by the aid of a gnomon 
of rude construction, having a dial divided into twelve parts. 



THE LOCAL EVENTS 

In the priestly calendar lore, magic and genuine science were inextricably 
entangled. The social necessity of measuring time arises from the seasonal 
fertility of man's biological allies, and the earliest explanations of the celestial 
events were frequently mixed up with man's preoccupation concerning his 
own fertility. What are sometimes offered as rival explanations of early 
practices are really different ways of saying the same thing. The phallic 



44 



Science for the Citizen 



tension of waking and the monthly cycle of a woman's life were closely 
associated with sunrise and lunar phenomena in the thought of primitive 
man. To say that an obelisk is a sundial, and to say that it is a phallic symbol 
involves no contradiction. Fertility and timekeeping were very closely con- 
nected in the same social context. Man had to be disciplined into the recog- 
nition that his own world is not the centre of the astronomical universe. He 
had to outgrow the belief that his own person is a sufficient model of natural 
processes in chemistry and biology. In psychology and social science he has 





FIG. 16 

Diagram to illustrate the slow retrograde motion of the moon's nodes. Two positions 
of the moon's orbit are shown, AB and CD, the latter position occurring about 4 or 5 
years after the former. Eclipses take place when, as at node C (solar) or D (lunar), the 
place where the moon's orbit crosses the plane of the earth's orbit is in line with the 
earth and the sun; provided, of course, that the moon is also at the node. (The tilt of 
the moon's orbit in relation to the ecliptic is here much exaggerated.) 

still to learn that individual preference is not a safe guide to the understanding 
of social behaviour. 

As liaison officers to the celestial beings, the priests found it paid to en- 
courage the belief that nature can be bought off with bribes like a big chief. 
One of their most powerful weapons was their ability to forecast eclipses 
(Figs. 17 and 18). Eclipses were indisputable signs of divine disapproval, 
and divine disapproval provided a cogent justification for raising the divine 
income tax. No practical utility other than the advancement of the priestly 
prestige and the wealth of the priesthood can account for the astonishingly 
painstaking attention paid to these phenomena. The moon's track lies very 
close to the ecliptic. If it moved exactly in the plane of the ecliptic, there 
would be a central eclipse of the sun every new moon, and a total eclipse of the 
moon every month, at the full. Careful measurement shows that its orbit is 



Pole Star and Pyramid 



45 




FIG. 17 



Total eclipse showing the sun's atmosphere just seen beyond the margin of the moon's 
disc (i.e. the sun's angular diameter is approximately the same as that of the moon). 




FIG. 18 

Eclipses. A. total solar eclipse, C. total lunar eclipse, B. earth's shadow seen as 
moon is at beginning of a lunar eclipse. 



46 Science for the Citizen 

inclined about 5 to the ecliptic (i.e. to the plane of the earth's orbit, as we 
now say). So the moon's path round the earth only cuts the earth's path 
round the sun (or the sun's apparent track round the earth) at two points 
called nodes (see p. 103), and an eclipse can only take place if the moon is at., 
or very near to, a node when the two nodes are in line with the sun and the 
earth. Relative to the fixed stars, the direction of the line which joins the 
nodes rotates slowly. The sun passes a particular node every 346-62 days. 
This is less than a year because the nodes are moving from east to west, 
and meet the sun before it completes its yearly circuit. So if earth, moon, 
nodes, and sun are in line at any time, they will be in line once more about 
eighteen years* later. More precisely this period is 18 years 11J days. If 
an eclipse occurs on a particular date somewhere on the earth's surface, 
another one will occur 18 years 11 J days later at a place about 120 W. on 
account of the odd third of a day. This cycle is still called the Saros, which 
is the name given to it by the Chaldean priests. It did not help people 
to arrange their meal-times and night journeys, to prepare for the lambing 
season, or to sow their crops. For the art of time reckoning the Saros had 
no particular use. Its discovery was prompted by a combination of super- 
stition and racketeering. Once made, it served to direct attention to two 
of the basic principles of scientific geography. Observation of eclipses in 
different places showed that solar time is local (see p. 78); and confirmed 
the belief that the earth is a spherical object (Fig. 18, B). The fact that 
lunar eclipses occur when the moon is practically in the ecliptic plane 
shows that the circular edge of the shadow on its face is the shadow of 
the earth itself. 

As soon as trade intercourse began, many other facts helped the growth 
of this belief. When ships appeared in the Mediterranean, maritime people 
became accustomed to the sight of the mast sinking last below the horizon, 
or the mountain rising first as land was sighted (Fig. 19). Little later than 
2000 B.C. Semitic traders were pushing north beyond the Mediterranean 
towards the Tin Isles, bringing back tales of the long summer days and the 
long winter nights of the northern regions. They told, too, how the aspect 
of the night sky changed. Stars low on the northern horizon became higher 
in the heavens as ships sailed into the northern seas. A fact most fatal to a 
flat earth view was that southern constellations disappeared entirely (Fig. 20) 
from view. To be sure, the belief that the earth is truly spherical (or nearly 
so) could only be settled by showing that a degree of latitude is the same 
distance if measured anywhere along any meridian of longitude, and a 
degree of longitude in the same latitude is always of the same length. We 
shall come to that later. Here it suffices to remark that the belief in the 
sphericity of the earth is of very great antiquity, and that there were a number 
of good reasons to support it. 

* 19 revolutions of the line of nodes relative to the sun take 346 -62 x 19 = 6585 78 
days. 223 lunar months = 6585-32 days. So that after 18 years 11 J days the line of 
node will have performed 19 revolutions relative to the sun and the moon will have 
performed 223 revolutions almost exactly. Hence the sun and the moon will occupy 
the same position relative to the nodes at the end of this period as at the beginning. 
How the position of the moon's node in any particular month is found is explained 
on p. 103. The rate of rotation is found by repeated observations of the same kind. 



Pole Star and Pyramid 



47 



FIRST STEPS TOWARDS A WORLD MAP 

The construction of the calendar taught mankind many lessons which 
were only fully assimilated when maritime trade developed. The phenomena 
on which we base the measurement of our fundamental units of time can 
only be explained if the spherical earth has a very definite orientation with 
respect to the celestial sphere. To begin with, we have already drawn the 
conclusion that there is one spot the North Pole which lies directly 
below the celestial pole or Pole Star. The plane of the ecliptic must cross 
the world at upper and lower limits defined by two belts where the sun will 



- observer J-Iori'zon lino. 



"" top (jf si ilp just sink- 
uio below /lonzori 
lin Q. 



BC= height: oi 

slup (h) 




FIG. 19. THE TANGENT OF THE HORIZON LINE 



be at its zenith (i.e. exactly overhead) on the summer and winter solstices 
respectively (Fig. 14). These are the tropics. Within the tropical belt the 
direction of the sun's noon shadow will change twice in the course of the 
year, a fact which could be observed in the southernmost parts of Egypt. At 
the equinoxes the sun occupies the apparent position where the ecliptic cuts 
the celestial plane at right angles to the axis of the celestial sphere. This 
plane, the celestial equator, cuts the earth at a belt, where day and night are 
always equal and the sun is at its zenith on the equinoxes. Finally, the axis 
at right angles to the ecliptic through the earth's centre must trace out a belt 
north of which the sun never sets at midsummer. Men were beginning to 
grasp these truths firmly long before they had first-hand experience of 



4 8 



Science for the Citizen 



them. In the fourth century B.C. the Greek materialist Bion taught the doctrine 
of the midnight sun in the Arctic Circle, which no ships had yet penetrated. 
The accuracy with which the Egyptian priests determined the length of 
the year, as well as the orientation of their temple sites, bear witness to 
something more than the recognition of an orderly sequence of events in the 
day and night sky. At an early date they had already begun to recognize certain 
metrical uniformities. Simple devices for measuring the angular bearings 
of a star are dated as early as 3000 B.C. The local bearings of a heavenly body 



J>ole 



I 




is seasonal , 
"*\ /since its lower 

$? transit is belcm^ 

x ^ the Jiortzon 

\ 

\ 

\ Observer \ 

\ I at mkktuAir \ 



tf 

/ Star B is 
,\ / seasonally visible 

\^^ (in 



FIG. 20. APPEARANCE OF THE NIGHT SKY IN DIFFERENT LATITUDES 
The sun is over the equator and the time of the year is supposed to be the autumn 
equinox. At latitude 30 star A will be visible throughout the night in autumn and 
invisible at midnight in spring. At latitude 60 it will be seen crossing the meridian 
above the pole just south of the zenith at midnight in autumn, and will be visible 
throughout the spring night below the pole, making its lower culmination at mid- 
night on the spring equinox. Star B will be visible at latitude 30 throughout part of 

at any hour can be fixed by two angles. One is its altitude (angular height 
above the horizon), or its complement called the zenith distance (Figs. 3, 21, 
and 114). The other bearing is its angular distance east or west of a fixed 
point on the horizon. Measured from the north point eastwards it is called 
the azimuth. For measuring altitude (or Z.D.) and azimuth a simple 
instrument essentially like the sort of astrolabes or quadrants used in ancient 
times can easily be made from a large protractor for blackboard use, as in 
Figs. 22 and 115. 
In taking the bearings of the stars in the night sky the two important 



Pole Star and Pyramid 



49 



principles which underlie the system of earth measurement by latitude and 
longitude were established long before the first star maps, described in the 
next chapter, were constructed. The first of these is implicit in the primitive 
art of telling the time by the stars. At a very early date the priestly custodians 
of the calendar were familiar with the use of simple mechanical devices for 
measuring short intervals of time. In principle they were analogous to the 
sand-glass which used to be sold to time the boiling of an egg. They could 
record the moment when some star crossed the meridian (i.e. when its 



^v, 

w 



Is circumpolar. 
leing vLsiile, at lower 
^T transit above the. lwr~ 
i7.on on Marcli 21 
at 



\ 



/ 



Observer 



\ 



ai muirugfar ^ 
Iat.60 ,6^)r;.2> \ 




visible. 



FIG. 20.Contd. 

the night during autumn a crossing the meridian towards the southern margin of the 
horizon at midnight on the equinox. It will be invisible in the spring night. At latitude 
60 it will never be visible, since its upper transit lies below the horizon plane. For 
the reasons explained on p. 58, with the aid of Figs. 29 and 30, 'the effective horizon 
may be drawn through the earth's centre parallel to the tangent of the horizon plane 
at the earth's surface. The part of the celestial sphere below the horizon is shaded. 



azimuth is or 180) and the interval which elapsed before another star 
crossed it by measuring the outflow of sand or water from a vessel. Hence 
they knew that the interval is constant for any two stars chosen. 

The stars appear to revolve uniformly around the pole, and their relative 
positions can be represented on semicircles radiating from the latter (see 
Fig. 48). The time which has elapsed since a star crossed the meridian can be 
measured by the angle through which its own celestial semicircle has rotated. 
Since the whole dome of heaven appears to rotate through 360 in 24 hours, 
an angle of 15, i.e. 360 24, between the celestial semicircles on which two 



50 Science for the Citizen 

stars lie is equivalent to a difference of an hour in their times of transit over 
the meridian. In the accompanying figure (Fig. 23) the star a in Cassiopeia 
(A) is roughly 30 from the furthermost of the five bright stars in the same 
constellation. So it takes about two hours for Cassiopeia to cross the meridian. 
The furthermost bright star (C) in the tail of the Great Bear is roughly 
210 from a Cassiopeiae. The difference in hour angles is 210 -~ 15 or 14 
hours. That is to say, the tail star of Ursa Major crosses the meridian at its 



noonsurL, 




FIG. 21 

The direction of a celestial body can be fixed by two angles, the angle it makes with 
the horizon or the vertical (altitude or zenith distance) and the angle it makes with 
the meridian (azimuth). 

"upper culmination/' roughly 14 hours after a Cassiopeiae. The hour angle 
differences of the stars give us an orderly picture of their appearance and 
disappearance on the horizon. The bright star Betelguese in the constellation 
of Orion is separated by an hour angle of nearly 6 hours from a Cassiopeiae. 
So when Cassiopeia is at its upper culmination, Betelguese will be rising 
almost exactly due east of the pole. See also Figs. 47, 48, and 49. 

Lockyer (Stonehenge and other Monuments) reproduces a figure illustrating 
a "night dial" used in fourteenth-century England to tell the time at night 
by watching the rotation of the two stars (pointers) of the Great Bear in 
line with the Pole Star. It is essentially a movable arm which rotates on 
a dial, the centre of which is perforated (see Fig. 147, Chapter V). To use 



Pole Star and Pyramid 51 

it, you would look at the Pole Star through this hole, move the dial in line 
with the pointers, and read off the time on the dial which could be cali- 
brated like a planisphere (p. 94) for each day in the year. Hooke (New 
Year's Day) states that pictorial representations of crude night dials based 
on essentially the same principle come from ancient Egypt, The exactitude 
with which the priests of these early civilizations endeavoured to determine 




FIG. 22 

A simple theodolite or astrolabe for measuring the angle a star (or any other object) 
makes with the horizon (altitude) or the vertical (zenith distance) can be made by 
fixing a piece of metal tube exactly parallel to the base line of a blackboard protractor, 
which you can buy from any educational dealer. To the centre point of the protractor 
fix a cord with a heavy weight (e.g. a lump of type metal which any compositor will 
give you free if you ask him nicely) to act as plumb line. The division opposite the 
cord when the object is sighted is its zenith distance (Z), and the altitude (h) is 90 Z. 
If you mount this to move freely on an upright wooden support which revolves freely 
on a base with a circular scale (made by screwing two protractors on to it) and fix a 
pointer in line with the tube, you can measure the azimuth (az) or bearing of a star 
or other object (e.g. the setting sun) from the north-south meridian. To do this fix 
the scale so that it reads when the sighter is pointed to the noon sun or the Pole 
Star. This was a type of instrument used to find latitude and longitude in the time of 
the Great Navigations. You can use it to find the latitude and longitude of your house, 
or to make an ordnance survey of your neighbourhood. 

the occurrence of celestial signals is illustrated by a Babylonian text which 
belongs to the time of Hammurabi. In it the heliacal risings of three con- 
stellations are given by weight (4 minas and 2 minas) corresponding to the 
outflow of water from a simple water clock or clepsydra analogous to the 
sandglass. 

The construction of the Great Pyramid shows that the Egyptians made 
exact measurements of the position of the stars at transit as well as when 
rising or setting. Such observations entail the recognition of a second 
uniformity of measurement. This also was a very early discovery. When it 
crosses the meridian, the altitude of any star (or its Z.D.) is always the same 
at a particular place. Since the pole is fixed, we may also represent the 



52 Science for the Citizen 

position of a star in the heavens by the difference between its meridian 
altitude* (or Z.D.) and the altitude (or Z.D.) of the Pole Star. This difference 
is now called the North Polar Distance, and its complement is now called 
the declination of the star. Thus: 

North Polar Distance = meridian altitude of star altitude of Pole 
or North Polar Distance = Z.D. of Pole Z.D. of star 
Declination = 90 - N.P.D. 

Declination is thus the elevation of a heavenly body above the celestial 
equator, and is of course negative in the case of bodies which transit south 
of the celestial equator (Fig. 24). 




(jreat Bear All 

FIG. 2;i THE STAR CLOCK 

Showing how the relative positions of the stars may be represented by the time 
interval between their transits over the meridian. The star A in Cassiopeia is almost 
directly above the pole, i.e. its hour angle is zero ( XXIV). The star C in the Dipper 
crossed the meridian ten hours earlier, i.e. its hour angle is X. 

NOTE. Since the celestial sphere rotates once in 24 hours the time which will 
elapse before a star crosses the meridian is the difference between 24 and its hour 
angle. Thus B in Cassiopeia will transit above the pole in 2 hours and its hour angle 
is 4- 22 or - 2. The hour angle of C in the Dipper is + 10 or - 14. 

The full significance of these facts did not clearly emerge until the study 
of the heavens received a new impetus in the great centres of Mediterranean 
shipping. As travel developed, the priestly lore of the calendar became the 
possession of Semitic trading peoples, who transmitted their knowledge to 
the Greek world. When astronomy ceased to be local and began to be 
international, observations made at different places could be compared. The 
result of this was another important discovery. Although the meridian 
altitudes vary from place to place, the N.P.D. of a star, like the difference 
in hour angle between any two stars, is the same everywhere (Fig. 24). Thus 
the position of the star in the heavens can be fixed for any observer by two 
coordinates its hour angle difference referred to any one star and its N.P.D. 
(or declination). These two data are all that is required for constructing a 

* Altitude, for this purpose, must be reckoned from the north point of the horizon. 
Zenith distances south of the zenith are reckoned as negative. 



Pole Star and Pyramid 



53 



star map or planisphere. From the star map it was a very short step to the 
first world maps of latitude and longitude. The Alexandrian maps opened 
up vistas of unexplored territory waiting for the ships of Columbus, 
Amerigo Vespucci, and Magellan. 



Difference, in 
altitude - 33 



Zenith 





difference, LTL 
- 33 



55 >2 a/>oa r <_' South 1 - 1 
Horizon A, 

/ / A '^ 



\ 



.Sourfieni / 
'Hori'zoti i 



/;/> 



FIG. 24 

The highest altitude of a star varies from place to place, but the difference between 
the highest altitudes of any two visible stars, i.e. the difference between their altitudes 
at meridian transit, is the same at all places. (The two stars chosen are Aldebaran 
and Sirius. They are drawn as if they crossed the celestial meridian at the same time. 
In reality Sirius transits about two hours later than Aldebaran.) 

One class of measurements of great antiquity arose in connexion with the 
fixing of the year from the summer solstices. This was the determination of 
the "obliquity of the ecliptic/' i.e. the angle which the sun's track in the 
zodiacal belt makes with the celestial equator. Chinese astronomers are said 
to have calculated the value correct within a quarter of a degree by as early 



54 



Science for the Citizen 



Summer Solstice 




, i/nrli 

IS 



/ 



=Zat.+ Cbtiquihr 

r ~r* T i 

or 



Sun's zenith, distance Suxis -zenith, distance. 

Ecliptic 

FIG. 25. EGYPTIAN MEASUREMENT OF THE OBLIQUITY OF THE ECLIPTIC FROM THE 

SUN'S NOON SHADOW 

At noon the sun is highest. The pole, the earth's centre, the observer, and the sun, 
are all in the same plane (or flat surface). On the equinoxes (March 21st and Sep- 
tember 23rd) the sun's zenith distance at noon is the observer's latitude (30 at 
Memphis). If the obliquity of the ecliptic is E 

L -f E = sun's zenith distance on winter solstice (December 21st). 

L E = sun's zenith distance on summer solstice (June 2 1st). 
So the obliquity of the ecliptic is 

(sun's z.d. on December 21st sun's z.d. on June 21st). 

as 1000 B.C. The Egyptians and Babylonians almost certainly had an equally 
accurate estimate at a much earlier date. The accuracy of these early estimates 
was achieved by the large scale of the instruments used, and gives us some 



Pole Star and Pyramid 55 

insight into the apparently pretentious dimensions of the great calendar 
monuments. Two obelisk measurements repeated over several years suffice 
(see Fig. 25) to give a very good mean value for the obliquity of the ecliptic. 
Unwittingly, as you will see from the same figure, those who first made these 
measurements also encountered another quantity. If Z w is the sun's zenith 
distance on the winter solstice and Z 8 is the sun's zenith distance on the 
summer solstice 



To them the quantity L was nothing more than the zenith distance (Fig. 26) 
of the noon sun on the equinoxes, hence a direct way of fixing the time of 



Zenith, 




FIG. 26. LATITUDE FROM SUN'S Z.D. AT NOON ON THE EQUINOXES 

On March 21st and September 23rd day and night are of equal length throughout the 
world. So the sun lies above the equator. At noon the sun always lies over the line 
joining the north and south points of the horizon, i.e. the observer's meridian of 
longitude. So the sun, the poles, observer, zenith, and earth centre are all in the same 
flat slab (or "plane") of space. Since the edges of a sunbeam are parallel, the sun's 
z.d. at noon on the equinoxes is the observer's latitude. 

a fertility festival which has persisted in the Lent observances of contemporary 
mythology. To us it is a ftindamental physical constant. It is numerically 
equivalent to what we now call the latitude of the observer. You can find the 
latitude of your house within a quarter of a degree by observing the sun's 
altitude at noon on March 21st or September 23rd. There will be a slight 
error in all such calculations if you only make the observation on one occasion, 
because it will rarely happen that the exact moment at which the sun is 
above the equator will also be exactly noon at the place where you live. 
The very precise measurements of the ancients represented the average of 



56 Science for the Citizen 

many observations made repeatedly. The ecliptic angle which defines the 
latitude of the tropical belts is between 23 and 24. At London (latitude 
51 N.) the sun's zenith distance at midsummer is 51 23- = 27 J% and 
at midwinter 51 + 23jl = 74|. The sun's altitude at noon is 62i 
(90 27J) on June 21st. The midwinter noon sun is only 15J (90 74) 
above the horizon. The Arctic Circle which defines the limit of the midnight 
sun on the summer solstice is at latitude 90 23i = 66| N. 



" PoU Star- 




Equator 



FIG. 27. LATITUDE FROM POLE STAR 

The altitude (horizon angle) of the Pole Star is the observer's latitude, both being 
equivalent to 90 z.d. of Pole Star. 

In making observations on the elevation of celestial bodies it is probable 
that the calendar makers of the early Mediterranean civilizations had dis- 
covered another fact of great geographical importance. On the equinoxes 
the sun's zenith distance is the same as the mean nightly altitude of the 
Pole Star. This was actually the star a in the constellation Draco at the 
time when the Great Pyramid was built. The meaning of this is easy to see. 
On the equinoxes the sun is in the plane of the celestial equator. The direc- 
tion of the Pole Star is at right angles to the celestial equator. You will 
therefore see (Fig. 27) that the elevation of the Pole Star above the horizon 
is the same as what we now call the latitude of the observer. In the northern 



Pole Star and Pyramid 57 

hemisphere all you have to do to find the latitude of your house is to take 
the horizon angle of the Pole Star with a home-made instrument like the one 
in Fig. 22. You should not make an error of more than 1. This tells you 
your distance from the equator or North Pole within 70 miles. 

Certain geometric principles underlie all these early measurements, and 
play an important part in the later developments of the Alexandrian astron- 
omy. Expressed in modern language, the first is that light rays from the 
heavenly bodies are parallel. At first sight this might seem a daring assump- 
tion. The fact is that the straight parallel edges of a beam of light were not 
laboratory curios to the first civilized men. In an age when there was no 
glass, and dust abounded, windows were high narrow slits to exclude wind 
and rain. The sharp straight edge of a shaft of bright light piercing a narrow 




FIG. 28 

The same star, whose declination is 45 North, as seen at transit from latitude 05 N., 
20 south of the zenith (z.d. 20) and from latitude 25 N., 20 north of the 
zenith (z.d. = + 20). 

orifice along a path of scintillating dust was one of the commonest facts of 
everyday life in ancient times. All the phenomena of shadow reckoning for 
timekeeping and architectural construction sustained the same belief. 

Then there was the principle of tangency demonstrated by the shadow 
cast by curved vessels. One of the earliest geometrical principles used in 
astronomy was that the line joining the zenith to the observer must pass 
through the earth's centre, if produced far enough. This follows from the 
elementary principle that the line joining the centre of a circle to the point 
where the tangent grazes the boundary is at right angles to it. Since the edges 
of the beam of light are straight, the observer's horizon plane is tangential to 
the earth's curved face. The zenith axis is the direction of the plumbline. 
The lines joining the zenith and the earth's centre to the observer both make 
right angles with his horizon plane, and are therefore collinear. The pole, 
the zenith, and observer are all in the same plane, which we have called the 
meridian. The earth's centre is in this plane, and so is the earth's axis, i.e. 



58 Science for the Citizen 

the line joining the pole to the earth's centre. This means that we can make 
fiat diagrams of the relations between an observer, a star, and the earth's 
centre, traced on sand or drawn on paper, provided that everything (Fig. 41) 
drawn is in the meridian plane at die same time. The direction of a star is 
given by the angle it makes with the plane of the equator at the earth's centre. 
As you will see from Figs. 28 and 31, this is the same as its declination, and 
explains why the N.P.D. (hence also declination) is independent of the 
observer's locality. 

Another simple geometrical principle which was implicit in the earliest 
attempts to diagrammatize the experiences which we have dealt with, is worth 
mentioning, because there is an inescapable distortion in any diagram like 
those in Fig. 20. The light beams of heavenly objects appear to be parallel 




FIG. 29 

If the size of the earth is very small compared with the distance of the stars, the 
observer's horizon practically coincides with a plane through the earth's centre parallel 
to the actual plane of the horizon. In other words, we see practically half the celestial 
sphere at any moment. 

because they are very far away. This means that the celestial sphere is 
immense compared with the earth. For practical purposes the observer's 
north-south horizon line can be replaced by a parallel line in the meridian 
plane drawn through the centre of the earth as in Figs. 29-30. So we actually 
see half the celestial sphere, and the east and west points of the horizon are 
in the same plane as the equator. 

Such conceptions did not grow out of a preconceived system of logic. 
It would be more correct to say that geometry grew out of them. Geometry 
was largely the offspring of astronomy and its sister craft of architecture in 
the calendar civilizations of the Mediterranean world. Admiration inspired 
by Greek literature has fostered a legend which associates Greek metaphysics 
with real or suppositious achievements of the Greeks in other domains. 
Although Euclid takes us through twelve books before he arrives at the 



Pole Star and Pyramid 



59 




----""" 

SOUTH [/"HORIZON" 
POINT * 



FIG. 30 

For the^ reason explained in^Fig. 29 the equinoctial or celestial equator cuts the 
horizon plane due east and due west. 



/ DeaitLa&on, circle of Star S 




ofC 



FIG. 31 

The direction of a star as seen from the earth is everywhere the same as measured by 
the angle between it and the celestial pole, i.e. the difference (N.P.D.) between its 
z.d. (or altitude) and that of the pole when it crosses the meridian. The N.P.D. is the 
complement of the angle (declination) which the star beam makes with the plane of 
the equator as it passes over the observer's meridian. The declination is equivalen t 
to the star's meridian zenith distance (E) as seen by an observer at the equator. 

N.P.D. = star's zenith distance at North Pole, 
Decl. star's zenith distance (E) at equator. 



60 Science for the Citizen 

area of the circle and sphere, the Egyptians already possessed correct 
formulae for the volume of a pyramid and the area of a sphere by about 
1500 B.C. (Moscow papyrus). The Rhind papyrus gives a value of TT equiva- 
lent to 3-16 in our notation, or less than 1 per cent out. Only gross over- 
valuation of the contents of books accounts for failure to appreciate the 
intellectual equipment of people who made the angles of the Great Pyramid 
with a precision equivalent to 0-06 of a degree. 



THE MODERN CALENDAR 

The publication of international radio programmes reminds us that our 
own civilization makes much more exacting demands on the precision of 
calendrical practice than did the seasonal practices of a primitive agrarian 
economy. To a large extent the elaboration of the principles involved in 
modern time-keeping has arisen from the needs of navigation and, in 
particular, from the problem of determining longitude with accuracy at sea. 
Hence the theoretical explanation of present-day practice must be deferred 
to another context, where the reader may meet it at a later stage in our 
narrative. The remainder of this chapter will be devoted to a brief survey 
of the evolution of timekeeping in historic times. 

There is no absolute measure of time. In the last resort all our mechanical 
instruments are checked against the same standard the interval between 
two meridian transits of the same celestial body. To put the matter in another 
way, the possibility of measuring time in shorter intervals rests solely on the 
fact that we are able to construct devices like hour-glasses which empty 
themselves, or pendulums which swing, a constant number of times between 
the appearance and re-appearance of the same star on the meridian. This 
fundamental unit is called the sidereal day. Its mechanical division into 
24 hours of 00 minutes and 3,600 seconds constitutes the actual system of 
time used in observatories in contradistinction to the system of time used 
in civil life. 

The apparent position of the sun in the ecliptic, when it is crossing the 
celestial equator at some time on the March equinox, is taken as the origin 
of reference in reckoning sidereal time. Sidereal time begins when the 
point where the celestial equator intersects the ecliptic crosses the meridian. 
This point of intersection, which is the Greenwich of the celestial sphere, 
is called the First Point of Aries, and is usually denoted by the ancient 
zodiacal sign for the ram (T). We shall see later (p. 95) how to find exactly 
where it is. From what has been said on page 50, you will see that the 
difference of hour angle between any star B and a star A on the celestial 
semicircle traced through T from pole to pole (see Figs. 23 and 49) is the 
sidereal time at which the star B appears on the meridian. The star a in 
Cassiopeia is very close to the hour circle of T , and since the sun is always 
less than a degree from T at noon on the vernal equinox, noon March 2lst 

/ 24 v 60 "N 

is within 4 minutes ( = r^ ) of zero time on the sidereal clock. So 
V^ ooO J 

the hour angle which separates a star from the first point of Aries is approxi- 



Pole Star and Pyramid 61 

* 

mately the ^ime wlfich elapses between its meridian transit and that of a 
Cassiopeiae. 

Although the sidereal day is the basic system of units and the one used 
in observatories, it is not suitable for everyday use. For everyday use the 
number assigned to an hour must correspond with our working, feeding, 
and sleeping habits. Everyday life was regulated by the solar day when 
sundials were used. The time of the day when the noon sun crosses 
the meridian., halfway between sunset and sunrise, always registers twelve 
o'clock on the sundial. Hour glasses or clocks, checked by the meridian 
transit of stars, register any number from to 24 sidereal hours at solar 
noons between one vernal equinox and another. This is because the solar 
day is longer than the sidereal day. The sun, as we have seen, is continually 
slipping backwards in the ecliptic. In a year it gets back to where it was 
before. It has therefore receded 360, which is the angle through which the 
stars rotate in one sidereal day. Thus the shadow clock loses one day per 
year. In an average day it loses 

24 'x 60 . . 

= 4 minutes (approximately) - 

ODO + 

The day after the vernal equinox solar noon would register hour 4 minutes 
instead of hour on the sidereal clock. Fifteen days after the vernal equinox 
noon would be 1 hour by the observatory clock. At the end of three months, 
i.e. on June 21st, noon would occur at 6 o'clock by sidereal time. 

The meridian transit of the sun gives us a solar day, which is approximately 
4 minutes longer than the sidereal day. Unfortunately the exact length of 
the solar day is not constant. A clock or hour glass which always registers 
the same number of sidereal hours and minutes between successive transits 
of the same fixed star does not register exactly the same number of minutes 
from noon to noon on different days of the year. The social inconvenience 
of this fact is what forces us to take the sidereal day as the fundamental 
unit to which everything else is referred. If we agree to regard the diurnal 
rotation of the fixed stars as our best possible approximation to an absolute 
standard, the relative irregularity of the solar day can be explained as the 
result of several circumstances of which the two most important will be 
explained in Chapter IV. They are the obliquity of the ecliptic, and the form of 
the earth's orbit. 

Owing to the obliquity of the ecliptic the sun's apparent annual motion is 
not in the same plane as its apparent diurnal rotation with the fixed stars. 
When it is "in Cancer" or "in Capricorn" it appears to be moving backwards 
in the ecliptic in a direction (Fig. 9) practically parallel to its apparent diurnal 
motion. In Aries or Libra its apparent path is very oblique. So even if the sun's 
apparent motion in the ecliptic were perfectly regular or, as we now say, even 
if the earth moved in its orbit at a uniform speed the shadow clock would not 
lose time at exactly the same rate throughout the seasons. Later we shall find 
that the angular velocity of the earth's orbital motion is not constant. Another 
fact, which did not emerge till it was possible to give a satisfactory account of 
the motions of the planets, is that the sun's apparent path is not exactly cir- 
cular, though nearly so. According to what we now know (Chapter IV) the 



62 Science for the Citizen 

earth moves in a nearly circular ellipse, and the sun is not exactly at its centre. 
Even if it moved with constant speed in its track, it would not move in equal 
periods through exactly the same angular distance from the sun. What is 
constant is its "areal velocity/' as Kepler discovered. 

In antiquity there was no correspondence between short intervals of 
astronomical and civil time. The civil day was reckoned from noon to noon 
by the shadow clock, and when weight-driven and spring clocks began to 
be used in mediaeval times they were continually corrected by reference to 
the sundial. The use of the solar day as a unit was abandoned when the 
determination of longitude at sea emerged in the practice of navigation. 
Solar time is local. That is to say, noon does not occur simultaneously at 
places on different meridians of longitude, and the determination of longitude 
in the daytime is made possible by this fact (see p. 78). To do this we need 
to know when it is noon at some standard meridian (that of Greenwich is 
generally accepted). This would be simple enough if an accurate clock set 
by Greenwich time would always record noon when noon occurs at Green- 
wich. Since the solar day is not a fixed number of pendulum swings or 
hair-spring oscillations, the unit of time taken is the mean length of the solar 
day throughout the year. On any particular day noon by mean time is a 
few minutes (never more than 18) before or after true solar noon. The differ- 
ence can be estimated by direct observation. It is tabulated in almanacs, 
being generally (though inappropriately) called the Equation of Time. The 
accompanying graph shows how it varies throughout the year (Fig. 32). 
It can be used to find sundial time at Greenwich if we have a reliable clock 
set to Greenwich mean time, and hence to compare sundial time at Greenwich 
with sundial time at any place to which the clock is taken. Likewise it can 
be used to set the clock to mean time by the sundial. 

Applying the equation of time may be illustrated by reference to a dis- 
crepancy which sometimes puzzles people when they consult almanacs. 
Times of sunset and sunrise are always given in mean time to meet the 
requirements of mechanical clocks. On this account the lengths of morning 
and afternoon may differ by as much as half an hour, or a little more. On 
September 23, 1935, Whitaker gives the times of sunrise as 5.48 a.m. and 
sunset as 17.57 (i.e. 5.57 p.m.). This would make morning reckoned from 
mean noon 12 5.48 = 6 hours 12 minutes, and afternoon 17.57 12.00 
= 5 hours 57 minutes, i.e. 15 minutes shorter than the morning. On the 
same day under "equation of time" we read "subtract from apparent time" 
(i.e. sundial time) "7 minutes 22 seconds." To the nearest half-minute 
therefore the mean time at true solar noon (which divides the day equally) 
was 11 hours 52| minutes a.m. This makes the times of morning (11 hours 
52 J minutes 5 hours 48 minutes) and afternoon (17 hours 57 minutes 
11 hours 52^ minutes) both the same, being 6 hours 4|- minutes on that 
day. For the convenience of mechanized transport clocks are regulated to keep 
mean time for some standard locality like Greenwich or Paris. This avoids 
recourse to elaborate arithmetical corrections for differences of local time, 
when time-tables are revised. So sundial time may differ considerably from 
the time of a clock set by radio signal, when the correction for longitude and 



Pole Star and Pyramid 63 

the equation of time are cumulative. Owing to the statutory introduction 
of Summer Time in England, an interval of about an hour and a half may 
separate the noon radio signal from sundial noon in Cornwall. 

The problem of choosing a socially convenient unit for short periods of 
time only became important when civilization spread into the comparatively 
sunless northern hemisphere and portable mechanical clocks replaced the 
sundial. A much earlier dilemma arose in accommodating astronomical 
observations to social convenience when fixing the length of the year. For 
civil use a year must consist of a whole number of days. By refined astro- 



"Mcdn, Tune 




FIG. 32. THE EQUATION OF TIME 

The Equation of Time is the misnomer for the number of minutes which, on any 
particular day of the year, must be added to true solar time ("apparent" time) to give 
mean time, or, conversely subtracted from mean time to give sundial time. It has 
four seasonal phases. The first extends from December 25th to April 16th, with 
its maximum about February 12th, when it reaches 14 minutes 23 seconds. During 
this period the equation of time is positive. Thus sundial noon at Greenwich on 
February 12th occurs at 12 hours 14 minutes 23 seconds G.M.T. In the next phase 
from April 16th to June 15th, it is negative. The maximum is 3 minutes 47 seconds 
about May 15th. Thus on that date sundial noon at Greenwich occurs at 11 hours 
66 minutes 13 seconds. From mid- June to September 1st the equation of time is 
again positive with a maximum of 6 minutes 22 seconds about July 25th. From 
September 1st till December 25th it is negative with a maximum about November 2nd, 
when it is necessary to subtract 16 minutes 22 seconds from sundial time to get mean 
time 

nomical measurement the year, as a record of seasonal events, is not an 
exact number of days. In order that the name or number assigned to a day 
may discharge its primary social use, it must keep pace with the seasons. 
Clearly it cannot do so if a year is always made up of the same number of days. 
From direct observations of the heavens we can define the year in two 
ways. The sidereal year is based on the same principle as the Egyptian year, 
which was announced by the heliacal rising of Sirius. It is the time taken 
by the sun to get back to precisely the same position in the heavens as some 
fixed stars. In contradistinction to the sidereal or Pyramid year, the Stone- 
henge year was solar, and corresponds to what is now called the tropical year. 
The tropical year is equivalent to taking the average number of days 
between noon at two successive vernal equinoxes. Astronomers define it 



64 Science for the Citizen 

as the time taken for the sun to get back to the First Point of Aries, i.e. where 
the sun is at some time before or after noon on the vernal equinox. On any 
particular occasion the interval between noon on the first and the succeeding 
vernal equinox is necessarily a whole number; but since the sun will not 
generally be exactly over the equator exactly at noon on any particular 
equinox where the observations are made, successive observations do not 
average out as a whole number. 

The average length of the solar or tropical year is not exactly the same as 
that of the sidereal year. The reason for this is a phenomenon known as the 
Precession of the Equinoxes. The Precession of the Equinoxes, which will be 
dealt with on page 219, is of great interest because it is one of the independent 
sources of evidence used to check the dates of historical events. The recur- 
rence of the seasons is due to the fact that the sun's apparent annual path 
is inclined to the earth's axis, or, alternatively, the earth moves round the 
sun in an orbit inclined to its axis which is always parallel in any two positions. 
Through the seasons the Pole Star retains its fixed position. Hence the earth's 
axis itself does not appreciably shift in a single year. On the other hand, 
comparison of astronomical observations on the declinations and hour angles 
of stars, carried out over long periods, shows that the plane of the celestial 
equator like that of the moon's orbit (Fig. 16) appears to rotate on the ecliptic 
at approximately the same inclination. Another way of saying this is that 
the plane of the terrestrial equator rotates on the earth's orbit (see Figs. 132 
and 133). One outstanding feature of this rotation is that the sun's position 
on the vernal equinox shifts round the ecliptic. In classical antiquity it 
occupied the same position in the heavens as the constellation of Aries. At 
the autumn equinox the sun was then in Libra. About 60 B.C. the sun's 
position at the equinoxes moved from Aries into Pisces and from Libra into 
Virgo. As a matter of fact we still speak of the First Point of Aries, though 
the sun is really in Pisces. On star globes the zodiacal signs, as used for 
months, no longer signify the actual constellation which is screened by the 
sun at that time. On December 21st the sun is now in Sagittarius and not 
in Capricorn; on June 21st it is in Gemini and not in the adjoining constel- 
lation of Cancer, as it was in ancient times. The phenomenon is generally 
said to have been discovered by Hipparchus, in 150 B.C., from comparison 
of earlier records with his own observations. According to Fotheringham's 
researches, it appears to have been elucidated at an earlier date (about 
340 B.C.) by a Babylonian astronomer, Cidenas. The passage of seventy 
years brings about a shift of 1 in the position of the equinoctial sun in the 
zodiacal belt. A complete rotation is a matter of about 26,000 years. 

Being at right angles to the plane of the equator the position of the celestial 
pole rotates very slowly round the pole of the ecliptic. We have spoken of the 
North Pole Star as if it were unchangeable and exactly located at the point 
round which the stars appear to revolve. It is a fortunate historical circumstance 
that there does happen to be a bright star practically at this point in our own 
time. There was also a bright star (a Draconis) very near it when the calendar 
cultures of the Nile and Mesopotamia flourished. Being fixed with reference 
to the earth the northern tunnel of the Great Pyramid always points near the 
celestial pole and not to any particular star that happens to be there. Today 



Pole Star and Pyramid 65 

it nearly points to another bright star round which different circumpolar 
constellations revolve. Our own Pole Star (Polaris in Ursa Minor) at present 
describes a minute circle almost exactly one degree off the pole. That is to say, 
the maximum difference of two determinations of its altitude at any place 
is 2. The average of two observations six months apart gives the correct 
position of the celestial pole, if we want an exact value for our latitude. On 
any one occasion we are not likely to be more than J out, an error of only 
35 miles, and therefore within sight of mountains on the sea horizon, when 
we are approaching land. There is no bright star near the south celestial 
pole. 

Between 2000 B.C. and about A.D. 1000 there was no very bright star near 
the pole. We might perhaps go so far as to say that the approach of Polaris 
to the celestial pole was the herald of the Great Navigations. 

Owing to precession the sidereal year gets out of step with the seasons in 
the course of centuries. The tropical year is about 20 minutes shorter than 
the sidereal year because the position of T changes at the rate of 1 degree 
in 70 years (precession of the equinoxes). In mean solar units its exact length 
is 365 days 5 hours 49 minutes to the nearest minute. For keeping track of the 
seasons the tropical year is the proper astronomical unit because it is based 
on a seasonal event. The length of the civil year is adjusted to correspond 
with the tropical year by periodic insertion of extra days in the leap years. 
The leap year system is analogous to an earlier practice, employed when the 
calendar still retained its lunar function. 

Since the Egyptians added five feast-days to twelve thirty-day months 
when they created the year of 365 days, it is probably that they first recognized 
a year of 360 days, or twelve 30-day months. There is evidence to show 
that the original length of the Chinese and the Sumerian year was also 
360 days. While the Egyptian calendar discarded any attempt to keep the 
divisions of the year in step with the phases of the moon, the oriental civil- 
izations were far less successful in devising a seasonal calendar. The mean 
length of the lunar month is almost exactly 29| days. To adjust the calendar 
to the phases of the moon the Babylonian priests introduced a regular 
alternation of 29- and 30-day months, with occasional intercalary months 
(or in our idiom "leap months"), to make up for the fact that the year of 
12 lunar months is only 354 days. To desert nomads the moon as a traveller's 
beacon has more significance than the celestial signals of a settled agronomy. 
This may be why the Mohammedan calendar retained the Babylonian year 
of 354 days without any intercalation, so that each month goes through all 
the seasons in a generation. The early Hindu astronomers introduced a 
regular five-year cycle, in which an extra month was inserted in the second 
and fifth years. Greek calendrical practice did not break away from the 
attempt to square lunar with solar time reckoning. It made a good com- 
promise by the introduction of a calendrical cycle due to Meton and Euctemon 
in 433 B.C. By intercalation of an extra month seven times in nineteen years, 
the beginning of a given named month did not suffer a seasonal shift of more 
than a day in two centuries. 

A great deal of confusion arose in the ancient chronologies from attempts 
to conserve the more primitive use of the lunar cycle as a time unit. Even 

c 



66 Science for the Citizen 

the Egyptian calendar must have been a month out after a lapse of about 
120 years. Petrie remarks that "the cycle of 1,460 years in which the calendar 
shifted round the seasons enables the record of any seasonal event to be a 
control on chronology/' and "the Berlin date" of 4241 B.C.,, which pushes 
back the origin of the Egyptian calendar to pre-dynastic times, is based on 
this fact. From texts extant in 2000 B.C. it appears that the Egyptian priests 
already recognized a "little year" or quadrennial cycle, analogous to our 
leap year, to compensate for the fact that their basic year of 365 days is roughly 
six hours too short. A brief and readable discussion of these early calendars is 
given in S. H. Hooke's book, New Year's Day. 

When Julius Caesar came into power he found the Roman calendar in a 
state of hopeless confusion. Alexandrian astronomy had then reached a high 
level of attainment. On the advice of the Alexandrian astronomer Sosigenes, 
and in accordance with his suggestions, Caesar established in 45 B.C. what is 
known as the Julian calendar. With a trifling modification, this continues 
in use among all civilized nations. The Julian calendar discarded all consider- 
ation of the moon, and adopting 365 \ days as the true length of the year, 
ordained that every fourth year should contain 360 days. The extra day was 
inserted by repeating the sixth day before the kalends of March (our 
February 24th). Later the extra day became our February 29th. Caesar 
also transferred the beginning of the year to January 1st. Up to that time it had 
been in March. This is still indicated by the names of several of the months, 
e.g. September, the seventh month, etc. Caesar took possession of the month 
Quintilis, naming it July after himself. His successor, Augustus, in a similar 
manner appropriated the next month, Sextilis, calling it August. 

The Alexandrian astronomers were not a priestly caste wedded to cere- 
monial practices. In acting on the advice of Sosigenes, Caesar took a step 
towards the secularization of the calendar. The complete secularization of the 
calendar did not come until after the invention of the clock. The next import- 
ant reform was adopted for sacerdotal reasons rather than scientific or 
utilitarian convenience, when a civilization equipped with mechanical clocks 
was already outgrowing the need for a religion of saints' days. The sunnier 
Moslem world encouraged astronomical studies as a secular branch of 
learning after the decay of the Alexandrian culture. In spreading over the 
cloudier northern hemisphere the Christian priesthood associated the seasonal 
festivals of barbarian tribes with a newly-acquired equipment of saints, who 
replaced the star god Pantheon when Christianity became the State religion 
of a collapsing empire. The monasteries which tolled the bell for vespers 
became what the Egyptian priesthood had been, the official timekeepers. 
They were the custodians of the hour candle, and they nursed the invention 
of the weight-driven clocks, which were put up in churches two centuries 
before they came into secular use. 

The fact that Christianity inescapably usurped the social function of the 
ancient priesthoods as the official timekeepers of civilization, when it became 
the imperial religion, explains the cordial encouragement which astronomy 
received from St. Augustine's teaching. Professor Farrington draws a sharp 
contrast between the words in which St. Augustine endorsed the position 
of astronomy in the curriculum of Christian education and his hostility to 



Pole Star and Pyramid 67 

other branches of pagan science. In de Doctrina Christiana St. Augustine 
says : 

A knowledge of the stars has a justification like that of history, in that from 
the present position and motion of the stars we can go back with certainty over 
their courses in the past. It enables us with equal certainty to look into the 
future, not with doubtful omens but on the basis of certain calculation, not 




FIG. 33. ALBRECHT DURER'S ADORATION OF THE LAMB 

In this woodcut from Durer's illustrations to the apocalypse (Die heimlich Offeribarung 
Johannis, 1498) the Paschal Lamb is superimposed on the noon sun in conformity to 
the astrological symbolism for the "Sun in Aries." The vernal equinox, when the sun 
enters Aries, was celebrated by fertility festivals in ancient times. In Christian ritual 
the Easter celebrations took their place, Easter being fixed with reference to the day 
when the sun enters the constellation of the Ram. 

to learn our own future, which is the crazy superstition of pagans, but so far 
as concerns the stars themselves. For just as one who observes the phases of 
the moon in its course when he has determined its size today, can tell you also 
its phase at a particular date in past years or in years to come, so with regard to 
every one of the stars those who observe them with knowledge can give equally 
certain answers. 



68 Science for the Citizen 

The last great reform was necessitated by the association of Easter cele- 
brations with the most ancient of the fertility festivals. Since the true length 
of the tropical year is a little less than 365 \ days (being 365 days 5 hours 
48 minutes 46 seconds, or a little over eleven minutes less), the Julian year 
is too long. By the year A.D. 1582 the error had assumed theological dimen- 
sions. The spring equinox had fallen back to the eleventh of March instead 
of falling on the twenty-first, as it did at the time of the Council of Nice, 
A.D. 325. Under the advice of the distinguished astronomer, Clavius, Pope 
Gregory, therefore, ordered the elimination of ten days. The day following 
October 4, 1582, was called the fifteenth instead of the fifth. To prevent 
further displacement of the equinox, it was also decreed that thereafter only 
such century years should be leap years as are divisible by 400. Thus, 1700, 
1800, 1900, 2100, and so on, are not leap years, while 1600 and 2000 are. 
The reformed calendar was immediately adopted by all Catholic countries, 
but the Greek Church and most Protestant nations refused to recognize the 
Pope's authority. In England it was finally adopted by an Act of Parliament 
passed in 1751. This provided that the year 1752 should begin on January 1st 
(instead of March 25th, as had long been the rule in England), and that the 
day following September 2, 1752, should be reckoned as the fourteenth 
instead of the third. There were riots in various parts of the country in 
consequence, especially at Bristol, where several persons were killed. Accord- 
ing to their slogan, they supposed that they had been robbed of eleven days, 
although the Act was carefully framed to prevent any injustice in the collec- 
tion of interest, the payment of rents, etc. The Julian calendar persisted in 
Russia until 1918, and in Rumania until 1919. When Alaska was taken over 
by the U.S.A. (1876) the official date had to be changed by only eleven days, 
one day being provided for in the alteration from the Asiatic reckoning to 
the American. 

Professor Elliot Smith's assertion that the magic of today is the discarded 
science of yesterday is well illustrated by an illuminating study of the role 
of the zodiacal constellation Cancer in ancient social practices (D'Arcy 
Thompson, Trans. Roy. Soc. Edinburgh, XXXIX). "We err in my opinion," 
says Professor D'Arcy Thompson, 

if we fail to recognize in this antiquated symbolism a deeper intention . . . 
I think we may discern that . . . these conventional but much varied collo- 
cations correspond in a singular and precise degree with natural groupings 
of constellations that are similarly figured and designated, and that the 
divinities with whom the emblems are associated had themselves a corre- 
sponding relation to the Signs of Heaven, where they had their places 
according to the doctrines of astrology, or which marked their festivals in the 
astronomic system of the sacred calendar. . . . We may abbreviate and sum- 
marize, as follows, the chief coincidences that have been related above : Cancer 
was domus Lunae, and the Crab is associated with the Moon on coins of 
Consentia, Terina, etc., with the lunar Diana of the Ephesians, and with 
various other images of the lunar goddess. Cancer was exaltatio Jovis, and 
the Crab is peculiarly associated with the Bird of Jove in the coinage of Agri- 
gentum, while the Aselli, individual stars of the same constellation, are mytho- 
logically associated with the same god. Cancer was sedes Mercurii, and the 
Crab is figured with the head of Hermes on coins of Aenus. Cancer rose with 



Pole Star and Pyramid 69 

Sirius, and there are dog-headed representations of both Mercury-Anubis and 
Luna-Hecate. Cancer marked the date of a festival of Pallas, and the Crab is 
figured with Pallas on coins of Cumae. Cancer is constellated with Hydra, 
simultaneously with part of which Virgo rose, and the Crab is associated 
with Hydra in the legend of Hercules, and hence with Lerna when the rites of 
the Virgo coelestis were performed. Cancer is in the neighbourhood of Corvus 
and Crater, and on coins of Mende the Ass is figured with the Raven and Cup. 
Cancer rose opposite to Aquila and Delphinus which set soon afterwards, and 
moreover, set precisely as the Dolphin rose; the Crab or the Ass is associated 
with the Eagle on coins of Agrigentum and Motya, and with the Dolphin on 
coins of Motya and Argolis. Cancer rose as Pegasus set, and the Crab is figured 
with the Horse on coins of Agrigentum. Cancer rose as the Southern Fish set, 
and the Crab and Ass are figured with the Fish on coins of Agrigentum and 
Cyzicus. 

MEGALITHIC REMAINS 

The conditions for fixing the year vary greatly in different parts of the 
world. Near the equator twilight is short, so that the interval between the 
visibility of a bright star and the moment of sunrise or sunset is small. On 
the other hand, the sun never rises or sets far north or south of the east and 
west point* Hence the determination of the year by heliacal rising or setting 
of bright stars is relatively easy, and the use of the solstice as a fixed point 
is relatively less convenient than it is in northerly latitudes like our own. 
At the latitude of Stonehenge twilight is prolonged, and the amplitude of 
the sun's shift between the solstices is large. On midsummer day it rises 
just over 40 north, and on December 21st 40 south of the east point, moving 
through an angle of over 80 in the course of the solstitial half-year. Hence 
it is not surprising to find evidence that megalithic monuments in northern 
Europe were set up to regulate a solar calendar. 

Sir Norman Lockyer, who carried out an extensive survey (Stonehenge 
and other British Stone Monuments} of such remains in Brittany, Wales, 
Cornwall, and Dartmoor, distinguishes between three types of solar calendar. 
One is the solstitial calendar based on the midsummer (June 2 1st- June 21st) 
or mistletoe (December 21st-December 21st) year. One is the equinoctial 
(spring or autumn) year. The other is the "vegetative" year. In early Greek 
and Latin calendars the quarter days (about May 6th, August 6th, Novem- 
ber 9th, and February 6th) are midway between a solstice and the ensuing 
equinox, and the beginning of the year was fixed in some of the Mediterranean 
cults at one of these dates, e.g. May 6th, by noting the position of the rising 
or setting sun when half the number of days (46) between the equinox and 
following solstice have elapsed. One advantage of a May (or August) year is 
that the same alignment serves to fix all four quarter days. If a line (Fig. 34) 
of stones is placed to greet the rising sun of June 21st, when the observer 
looks northward, it will also greet the setting sun of December 21st, when 
the observer looks southward. If a line is placed to greet the rising sun of 
May 6th, when the sun is still travelling northward, it will also greet the 
rising sun of August 6th on its return journey. Similarly, if the observer 
looks towards the southern horizon, it will equally well serve to fix the 
position of sunset on November 6th and February 6th. In northerly latitudes 



yo 



Science for the Citizen 



sErriKt* 
TVT 




FIG. 34 

Diagram to show how a double circle of stones arranged to sight the rising and setting 
of stars can be used to keep track of the seasons. The directions and times are cal- 
culated for today at about the latitude of Stonehenge where Vega just dips beneath 
the northern horizon and the star Capella never sinks below it. Vega setting in evening 
twilight in early March and rising in evening twilight in early April announces the 
coming and going of the vernal equinox, when Capella is seen skirting 7 above the 
north point of the horizon at lower transit in morning twilight. Castor setting in 
morning twilight in early March also warns the coming of the vernal equinox. Sirius 
rising, Altair and Fomalhaut setting in morning twilight in late August anticipate the 
onset of the autumnal equinox. The approach of midsummer day is announced by 
the Pleiades rising and Antares setting in morning twilight at the end of May. Mid- 
winter is announced by Aldebaran setting in morning twilight in late November 
and the setting of the Pleiades and the belt of Orion with the rising of Antares in 
morning twilight in early December. Fomalhaut skirting the south point of the 
horizon in evening twilight about January 1st announces that the solstice is past. 
The November quarter-day is announced by Deneb skirting 5 above the north 
point of the horizon at lower transit in morning twilight. 

where the sun is often obscured by rain or mist on any particular day of the 
year the advantages of this are obvious. 

The megalithic relics of Northern Europe (stone circles, avenues or rows, 
monoliths or menhirs, dolmens) have remarkably constant features. From 
a study of the alignment of stone rows and avenues or neighbouring menhirs, 



Pole Star and Pyramid 71 

Lockyer found that a very large proportion are placed to greet the rising 
sun about May 6th, and, allowing for the slight change in the obliquity of 
the ecliptic in the course of the last five thousand years, he dates them about 
1 500 B.C. Associated with the solar alignments there are often found to be 
others which correspond to the rising or setting position of a bright star 
whose heliacal rising or setting at this date acted as a warning of the approach 
of the May new year's day. Since there are comparatively few bright stars 
which could serve the purpose, the established use of the same ones as signals 
of solar events in the Mediterranean cults leaves little doubt that the stone 
circle was a calendrical observatory, and the row or avenue leading to one 
was a via sacra for the ritual procession to watch the rising sun of the quarter 
day, or the heliacal rising and setting of the star (Pleiades, Sirius, Arcturus, 
etc.) which was the herald of its approach. 

EXAMPLES ON CHAPTER I 

1. With the aid of diagrams compare the behaviour of the sun's shadow at 
latitudes 70 N., 50 N., 10 N., and 0, 10 S 50 S., 70 S. throughout the 
seasons. 

2. The star a in the constellation Ursa Major was seen at its lower culmina- 
tion at midnight on September 4th. Later on it was seen due east of the Pole 
at 2 a.m. What was the date? 

3. Given the obliquity of the ecliptic as 23 J, find the sun's zenith distance 
on December 21st at a place where the altitude of the Polar Star is 51. 

4. The altitude of the Pole Star at a certain place is found to be 53. Find 
the approximate date on which the sun's noon altitude was 60, given the fact 
that the sun rose earlier on the previous day. 

5. Imagine that you have been deported to an unknown destination at which 
you arrive after months of high fever. How would you be able to recognize 

(a) whether you were near the North Pole, the equator, the South Pole, the 
tropical belts; (b) how would you be able to locate your approximate position 
between these limits without any instruments; (c) how would you determine 
roughly the time of year? 

6. What additional information could you gain, if you had a protractor, some 
string, and a few nails? 

7. Suppose yourself in the same situation. You only recall the following 
geometrical theorems of your schooldays: (i) The theorem of Pythagoras, (ii) the 
angles at the base of an isosceles triangle are equal, (iii) the angle at the centre 
of a circle is twice the angle at the circumference subtending the same arc. 
How could you use these to make angles of 90, 45, 30, and 60 degrees round a 
shadow pole with a piece of cord and a peg? How would you proceed to obtain 
other angular divisions? 

8. Assuming that you have done this, how would you find (a) your latitude, 

(b) the date, from observations of the sun's noon shadow? 

9. How would you make an hour glass to record the time elapsing between 
the meridian transits of the stars? 



72 Science for the Citizen 

10. Explain why leap years are inserted and what is meant by Greenwich 
mean time. 

11. If Sirius transits 23 above the southern horizon at lat. 60 N., show 
with a diagram like that of Fig. 24 at what latitude it will 

(a) just graze the southern horizon at transit; 

(b) transit 16J south of the zenith. 

12. If Aldebaran transits 12| south of the zenith at latitude 29 N., at what 
latitude will it 

(a) transit exactly overhead? 

(&) transit 5 north of the zenith? 

(c) just graze the northern horizon as a circumpolar star at lower 

culmination? 

Test the rule that if the altitude is always measured from the northern horizon, 
the formula 

altitude = 90 (zenith distance) 

still holds good, if the zenith distance is negative, when a star transits south of 
the zenith. 

13. Find the north polar distance of Aldebaran and the south polar distance 
of Sirius with diagrams like those in Fig. 24. Hence find with similar figures 
at what southern latitude 

(a) Aldebaran ceases to be visible at all. 

(b) Sirius becomes a circumpolar star of the south celestial hemisphere . 

14. Find at transit the zenith distance of Sirius and Aldebaran at latitude 
5 N. and latitude 5 S. and of Sirius at latitude 22| S. Show that if (a) z.d. 
south of the zenith is negative, (b) declination south of the equator is negative, 
(c) latitude south of the equator is negative, the following formula applies to all 
situations : 

Declination = Latitude -j- zenith distance. 

15. Draw a figure to show the upper and lower transit of Sirius at 85 S. 
Show that in the southern hemisphere, where there is no bright star near the 
celestial pole, the latitude, i.e. elevation of the pole, is the mean altitude of a 
circumpolar star at upper and lower culmination reckoned from the southern 
horizon. 

16. If solar time at Exeter is 14 minutes behind solar time at Greenwich, on 
what days of the year (see Fig. 32) will 

(a) noon on the sundial at Exeter agree with broadcast time (G.M.T.) 

from Greenwich? 
(6) noon on the sundial occur two minutes after broadcast noon? 

(c) noon on the sundial occur twenty minutes after broadcast noon? 

17. On November 5th the sun crosses the meridian at one o'clock G.M.T. 
At what time by a chronometer set to G.M.T. will it cross the meridian at the 
same place on February 5th, June 21st, and August 4th? 



CHAPTER II 

POMPEY'S PILLAR 

The Science of Seafaring 

IN the everyday life of mankind the first fact which led to the growth of an 
organized body of scientific knowledge was the fertility of the crops and 
flocks. The second was the freedom of the seas. From the first came the 
need for an organized calendar, from the second a system of earth survey. 
The following passage from Nilsson's monograph illustrates the same social 
needs at work in the culture of backward communities today : 

The calendar and practical life become to some degree separated from 
each other; the first lays the principal emphasis upon the correct ordering of 
the series of days, which is of especial importance on religious grounds for the 
selection of days and the fixing of the right day for the religious observances; 
in practical life, however, the point of chief importance is to determine the 
times when the various occupations may be begun and sea-voyages undertaken, 
both of which depend upon the solar year, and for this the stars afford the best 
aid. Hence it happens that sometimes the reckoning by the stars appears, as 
one more profanely determined, in a certain opposition to the lunisolar reckon- 
ing, which has a more religious character. This happened in ancient Greece, 
where the stars served for the time-reckoning of sailors and peasants while the 
lunisolar calendar was developed and extended under sacral influence; the 
festival calendar, which was regulated and recorded by the moon, became 
the official civil calendar. It was only later that the stellar calendar was sys- 
tematically brought under the influence of the fully developed astronomy and 
of the Julian calendar. In sailing, the stars afford to the primitive seafaring 
peoples the only means of finding their way when the land can no longer be 
seen. From the necessities of seafaring the greatly advanced knowledge of the 
stars possessed by the South Sea peoples has arisen; this is because practical 
ends are served not by a priestly wisdom, but by a profane. Nevertheless the 
knowledge of the stars is a secret which is carefully guarded in certain families, 
and kept from the common people as is reported of the Marshall Islands. 
Among the Moanu of the Admiralty Islands it is the chiefs who are initiated 
by tradition into the science of the stars. On the Mortlock Islands, where the 
science of the stars is very highly developed, there was a special astronomical 
profession; the knowledge of the stars was a source of respect and influence, it 
was anxiously concealed, and only communicated to specially chosen indi- 
viduals. Only a few can determine the hours of night by the stars. The Tahitian 
Tupaya, who accompanied Cook on his first voyage, was a man of this kind, 
specially distinguished for his nautical knowledge of the stars. The elements 
of the science, however, seem to have been pretty generally known, and from 
the Caroline Islands comes a curious account of a general instruction therein. 
It was first mentioned by the Spanish missionary Cantova in the year 1721, 
and was later confirmed by Arago. In every settlement there were two houses, 
in one of which the boys were instructed in the knowledge of the stars, and in 
the other the girls; only vague ideas were imparted, however. The teacher 



74 Science for the Citizen 

had a kind of globe of the heavens on which the principal stars were marked, 
and he pointed out to his pupils the direction which they must follow on their 
various journeys. One native could also represent on a table by means of grains 
of maize the constellations known to him. This is a nautical non-priestly 
astronomy, which has really little to do with calendrical matters in general, 
although as a matter of fact in the Carolines and the Mortlock Islands it has 
led to the naming of all months from constellations, and therefore to a systematic 
sidereal regulation of the calendar. 

In ancient times transport by land involved a large initial outlay in making 
roads and a vast output of energy in carrying loads. Shipping dispensed with 
the need for roads and registered man's first tentative exploit in replacing 
the effort of man or beast by the forces of inanimate nature. The Semitic 
peoples of Asia Minor had established colonies throughout the Mediterranean 
world by the beginning of the second millennium B.C. Carthage was founded 
in the ninth century. By then ships had begun to track north towards the 
Tin Isles of the north. In the sixth century Carthaginian navigators had coasted 
along West Africa beyond the equator. From these Semitic peoples of Asia 
Minor the maritime Greeks of the seventh and sixth centuries B.C. absorbed 
the star lore of the ancient world and the geometrical principles of the 
calendar architecture. Their first two teachers of repute Thales and 
Pythagoras were both of Phoenician parentage. So was Anaxagoras, who 
brought astronomy from Miletus to the Court of Pericles. Pythagoras is said 
to have acquired in Egypt his knowledge of the obliquity of the ecliptic and 
the identity of the planets Mercury and Venus in their appearances as 
morning and as evening stars (see p. 184). According to Callimachus (cited by 
Diogenes Laertius), Thales determined the position of the stars in the Little 
Bear by which the Phoenicians guided themselves in their voyages. From 
Professor Taylor's studies on mediaeval pilot books it seems that a similar 
practice persisted in the Christian era. Fig. 67 shows how the relative positions 
of stars in the Little Bear were once used to estimate the error involved in 
determining latitude by the Pole Star, when it was not so near the true centre 
of the heavens as it is now. The orientation of two stars in the Little Bear 
above, below, east, or west of the true pole made it possible to apply the 
necessary correction, ranging from 3:> to + 3| in the fifteenth century. 

Scientific geography starts with the exploration of the Mediterranean, 
and the determination of latitude, which probably antedates Greek civiliza- 
tion, arose by easy steps from the practice of navigating by the Pole Star. 
That maritime astronomy had reached a high level of precision before the 
great advances made by the Alexandrians is indicated by the reference to 
Pytheas in the following passage from Professor Farrington's Science in 
Antiquity : 

. . . about 500 B.C. the Carthaginian, Hanno, coasted down the west coast 
of Africa as far as Sierra Leone, to within 8 degrees of the equator. Close on 
the heels of the Phoenicians followed their rivals, the Greeks. The effective 
discovery of the Black Sea was the work of the Greeks of Miletus. Exploration 
was begun about 800 B.C., and by 650 B.C. there was a heavy fringe of Greek 
colonies all round the Black Sea coast. It was for this city of explorers and 
merchants that Anaximander constructed his map. It was not from Miletus, 



Pompey's Pillar 



75 




MAP OF THE WORLD BY HECATAEUS, 517 B.C Showing the primitive ideas held at 
the time of Pythagoras. From Breasted's Ancient Times. 



*--* --=r.-,' C*"~' r 

:^^^m. _, , 




MAP OF THE WORLD ACCORDING TO ERATOSTHENES, ABOUT 250 B.C. 




PTOLEMY'S MAP OF THE WORLD, ABOUT A.D. 200 
FIG. 35 



j6 Science for the Citizen 

however, but from the neighbouring town of Phocaea, that the great enterprise 
was organized that eventually turned the Mediterranean into something like 
a Greek lake. Probably as early as 850 B.C. colonists from Phocaea occupied the 
Italian site of Cumae near Naples, and by the end of the seventh century 
numerous Greek settlements, mostly Phocaean in origin, dotted the western 
Mediterranean. The most westerly of these settlements was at Maenace, near 
Malaga in Spain, the most famous was at the first site of civilization in France, 
Marseilles. It was from Marseilles that a Phocaean sea-captain, Pytheas, about 
300 B.C., made one of the great voyages of antiquity. Eluding the vigilance of 
the Carthaginians, who still at that date endeavoured to keep the Atlantic as 
their preserve, he slipped through the Pillars of Hercules and coasted north. 
His chief objective was the tin mines of Cornwall, which he visited and well 
describes. Pytheas was an educated man and a skilled astronomer, and his 
voyage was rich in scientific results. He was capable of discovering that the 
Pole Star is not situated exactly at the pole, and of determining the latitude 
of Marseilles to within a few minutes of the correct figure. His accurate obser- 
vations gave later geographers their reference points in plotting the map of 
northern and central Europe. . . . Scientific voyagers like Pytheas were 
capable of calculating latitudes astronomically. Longitude, however, remained 
a matter of dead-reckoning. 

The positive contributions of the Greeks to the advancement of new 
knowledge of nature are easy to exaggerate, and what useful discoveries they 
made belong to the earlier period and localities where they were in closest 
touch with the great trading centres of northern Palestine. It is only fair 
to say that the Ionian Greeks never made the claims to originality ascribed 
to their Attic successors by the lexicographers of later time. Democritus, the 
doyen of Greek materialism, has left us the following fragment which is 
eloquent of their debt to the past: 

Of all my contemporaries it is I who have traversed the greater part of the 
earth, visited the most distant regions, studied climates the most diverse, 
countries the most varied, and listened to the most men. There is no one who 
has surpassed me in geometrical constructions and demonstrations, not even 
the geometers of Egypt, among whom I passed five full years of my life. 

The great contribution of the Greeks to the advance of knowledge was 
that they took a decisive step away from the association of natural enquiry 
with ritual. They travelled. They were curious, and they recorded the results 
of their travels in a language better fitted to precise description than the 
scripts of the priesthood. Since it is always easier to separate what is mere 
magic from what is real science in the culture of foreigners, they were able 
to discard some of the superstitions of their schoolmasters. Eventually they 
fell victims to a practice which is no less pernicious than priesthood itself. 
As time passed the pursuit of science and the art of mathematics which 
should be its servant came to be regarded more and more as mere recreation 
for a leisured and litigious class. They developed that inveterate belief in 
logic which is the password of the legal profession and the peril of science. 
Mathematics became the master instead of the servant, and was itself sterilized 
by losing contact with the world's work. 

It is sometimes suggested that naturalism and political philosophy 



Pompey's Pillar 77 

flourished side by side as twin growths from the same soil of intellectual 
freedom. To a large extent they were chronologically and topographically 
separate. Aristotle, whose person symbolizes the fusion of positive science 
and political speculation, devoted much of his energy to verbal refutations 
of the sound experimental physics of his Ionian predecessors. Plato, who 
declared that all the works of Democritus ought to be burned, did not 
shrink from contemptuous references to astronomical observations or to the 
pioneer acoustical experiments of the Pythagoreans. The following (Republic, 
Book VII) is typical of the anti-scientific temper of Plato's teaching:* 

It makes no difference whether a person stares stupidly at the sky, or looks 
with half-shut eyes upon the ground; so long as he is trying to study any sensible 
object, I deny that he can ever be said to have learned anything, because no 
objects of sense admit the scientific treatment. . . . Do you not think that the 
genuine astronomer will . . . regard the heaven itself as framed by the 
heavenly architect? . . . But as to the proportion which the day bears to the 
night, both to the month, the month to the year, and the other stars to the sun 
and the moon, and to one another will he not, think you, look down upon the 
man who believes such corporeal and visible objects to be changeless and exempt 
from all perturbations; and will he not hold it absurd to bestow extraordinary 
pains on the endeavour to apprehend their true relations? 

We shall pursue astronomy with the help of problems just as we pursue 
geometry but we shall let the heavenly bodies alone, if it is our design to 
become really acquainted with astronomy and by that means to convert the 
natural intelligence of the soul from a useless into a useful possession. . . . 
You can scarcely be ignorant that harmony also is treated just like astronomy 
in this, that its professors like the astronomers are content to measure the notes 
and concords distinguished by the ear one against another, and therefore toil 
without result. Yes, indeed, they make themselves ridiculous. They talk about 
"repetitions" and apply their ears closely, as if they were bent on extracting 
a note from their neighbours; and then one party asserts that an intermediate 
sound can still be detected, which is the smallest interval, and ought to be the 
unit of measure; while the other party contends that now the sounds are 
identical both alike postponing their reason to their ears. . . . They act like 
the astronomers, that is, they investigate the numerical relations subsisting 
between these audible concords, but they refuse to apply themselves to problems 
with the object of examining what numbers are, and what numbers are not, 
consonant, and what is the reason of the difference ... a work useful in the 
search after the beautiful and the good, though useless if pursued with 
other ends. 

From this morass of metaphysical disputation, into which the secular 
knowledge received by the Attic Greeks from their Ionian predecessors 
had fallen, the culture of classical antiquity was rescued by the brutal realities 
of war. The conquests of Alexander brought new opportunities of intellectual 
intercourse, new contacts with the world's affairs, and led to the foundation 
of a cosmopolitan centre of learning near the site of the greatest Egyptian 
triumphs of architecture and irrigation. Alexandria became a foremost centre 
of Mediterranean shipping. The daily record of a ship's progress which is 

* My friend Professor Ferguson thinks that Plato's later disciples and commentators 
misinterpreted this passage, which was really a plea for hypothesis. 



78 Science for the Citizen 

put up in the saloon of a modern liner is part of the cultural debt which we 
owe to the brilliant efflorescence of discoveries made in Alexandria during 
the three centuries which preceded the beginning of the Christian era, and 
the three centuries before Rome licensed the monks to loot the treasury of 
pagan science. 

We have seen that the cardinal achievement of the priestly culture of the 
great calendar civilizations was the discovery of the year. The determination 
of the year entailed careful daily measurement of the sun's shadow and the 
bearings of the stars in the night sky. In making these measurements men 
blundered into the recognition of certain physical constants which provide 
a scientific basis for fixing our position in the universe of stars and upon 
the earth's surface. Three fundamental relations to which we have already 
directed attention are the altitude of the pole, the tilt of the plane of the sun's 
apparent track on the equinoxes, and the inclination of the ecliptic to the 
equator. The knowledge of these relations gives us all that we require to 
estimate the northerly or southerly position of an observer with reference to 
the poles or the equator, and divides the earth into three zones with charac- 
teristic seasonal phenomena on either side of the equator. These are the 
frigid (Arctic or Antarctic), temperate, and tropical regions. A fourth 
uniformity, the constant difference of hour angle between the transit of two 
stars, gives us some of the information from which we can determine our 
position east or west of any fixed line from pole to pole. 

The full significance of none of these relations could emerge until it was 
possible to draw comparisons between observations made at different places 
and to sift what was purely local information about shadow, star bearings, 
and seasons, from facts which are common ground to people inhabiting 
different parts of the world. Without doubt by far the most difficult step in 
this advance was to understand the geographical meaning of the time relations 
between the transits of heavenly bodies. Today it is easy enough to establish 
the fact that astronomical, e.g. solar, time is local. There is no difficulty in 
determining within a minute the exact time at which the sun crosses the 
meridian by means of a home-made apparatus like that shown in Fig. 22. 
Noon for you will be the time when the zenith distance of the sun is least. 
On four days of the year (April 15th, June 15th, September 1st, and Decem- 
ber 25th see Fig. 31) the correction (called the "equation of time") necessary 
to make the length of the solar day bear a constant relation to the interval 
between successive transits of the same star, vanishes. That is to say, 12 noon 
G.M.T. is the true solar noon at Greenwich. If you live in Cornwall, you 
will find that noon by your own sundial does not agree with a watch set 
by the daily broadcast announcement of Greenwich time. Near Land's End 
on these days noon will occur when the watch records the time as 12.23. 

The explanation of this fact is one which we generally learn in our school- 
days. It is shown in Fig. 36. The earth rotates eastward to meet the rising 
sun. If we are west of another place, we have farther to travel before the sun 
crosses our meridian. The earth revolves through 360 in 24 hours, i.e. 15 
per hour, or 1 in 4 minutes. Hence, time is local reckoned from the transit 
of a celestial body, whether it is sidereal time (reckoned from when T crosses 
our meridian) or solar time (reckoned from when the sun passes over our 



Pompey's Pillar 



79 



meridian). That local noon is behind noon at another place., e.g. Greenwich, 
means that when it is noon at the latter we have still to go some distance 
before our meridian reaches the sun or alternatively the sun has still to 
travel westwards some distance to reach our meridian. In other words, we 
are west of it. If local noon occurs 23 minutes after Greenwich noon, the 

/23\ 
earth has to rotate eastwards through an angle of (-j-) 5f to bring the 

sun to our meridian after crossing the meridian at Greenwich. This angle is 
our west longitude. So we have the following simple rule for determining 







about the 
Polar Axis 
o 




30E (2p 777 ) 
30V 60E.(4p.rn : ) 



FIG. 36. FINDING LONGITUDE 

At noon the sun lies directly over the line joining the north and south points of the 
horizon, i.e. the meridian of longitude on which you are located. In the figure it is 
directly above the Greenwich meridian,, and it is therefore noon at Greenwich. If 
you are 30 East of Greenwich the earth has rotated through 30 since your sundial 
registered noon. It has therefore made one-twelfth of its twenty- four-hourly revolution, 
so that it is now two o'clock by the sundial. If you are 60 West the earth has still to 
rotate 60 before the sun will be over your meridian, i.e. one-sixth of its twenty-four- 
hourly rotation; so your sundial will register 8 a.m. 

longitude. If noon at a place B happens x minutes or y hours after noon 

x 
at A, B is j or 15jy degrees west of A, and if noon happens at B x minutes 

x 
ory hours before A, B is - or \&y degrees east of A. Difference in longitude 

is the hour angle difference of the local transit of the sun across the meridian. 
Simple as this is, it was a very difficult step to make in ancient times. 
Until there were wheel-driven clocks, there were no portable devices with 
which to compare solar time at two different places. Although the later 
Alexandrians had very elaborate water-clocks of much greater delicacy 
than the crude hour-glasses of an earlier period, and certainly no less 
accurate thai* the first mediaeval clocks which were driven by weights, 



8o 



Science for the Citizen 



they had no means of maintaining a continuous record of time over a 
long journey. The way in which they escaped from this limitation is 
not difficult to reconstruct, because we know the methods which were first 
used to determine terrestrial longitude. Although many of the hypotheses 
framed by the first civilized men about the influence of the heavenly bodies 
are now discarded as magical, they led people to observe phenomena which 
proved to be of great use to their successors in a different social setting. 
In particular the study of eclipses as events of august omen focussed close 
attention on the behaviour of the moon. Aside from eclipses, the moon's 
course displayed other circumstances which were highly portentous. As it 
moves through its orbit, it may intercept the visibility of other heavenly 
bodies, in particular the planets which move near the plane of the ecliptic 
and the moon's orbit. Thus there will be an occultation of Mars, i.e. Mars 




VI a.m- 
FIG. 37. FINDING LONGITUDE BY A LUNAR ECLIPSE 

The figure is drawn like Fig. 36 with the South Pole nearest to the observer, so that 
you see the meridians of the places named. The places themselves, being in the 
northern hemisphere, are not seen. 

will disappear behind the moon's disc, from time to time, just as at times 
the sun would seem to move behind the moon's disc in a solar eclipse. The 
care bestowed on the study of eclipses and occupations has outlived con- 
fidence in the political and personal effects attributed to them. So also the 
usefulness of observations with the spectroscope may well outlast the wave 
theory of light and the ether. People observed them and were able to know 
when they would occur at a particular place. Anyone who possessed an 
hour-glass at another place could watch for them and record the interval 
which elapsed between an eclipse or occultation and the foregoing noon. 

The illustration in Fig. 37 will make this clear. If a lunar eclipse is seen 
at 5.58 p.m. by Greenwich local time, you would see the moon's disc entering 
the earth's shadow cone at Aden just before nine o'clock (8.58) according 
to a watch set by the Aden sundial. You would therefore be (8.58 5.58)=^ 3 
hours in front of Greenwich time, and therefore you could conclude that 
Aden is 3 x 15 = 45 East of Greenwich. The mariner who possesses an 
almanac giving the local time at which an eclipse or other celestial signal 



Pompey's Pillar 81 

will occur at one place., can therefore obtain his longitude by recording the 
local time of its occurrence where he is. According to Marguet (Histoire de 
la Longitude de la Mer au XVIII siecle en France}, Columbus looked out for 
a port of anchor at Haiti for observing at rest the conjunction of sun and 
moon on January 13, 1493. Bensaude (UAstronomie Nautique au Portugal 
a Vepoque des grandes decouvertes) tells us that Amerigo Vespucci found the 
difference of longitude between Venezuela and Cadiz in 1499 by observation 
of a lunar eclipse. 

In his book The Geographical Lore of the Time of the Crusades, J. K. Wright 
tells us how the Alexandrians, equipped with neither chronometers nor 
wireless signals, relied on this method of finding longitude to construct the 
first true maps : 

Eratosthenes, Hipparchus, Pliny, and Ptolemy all understood that it may be 
found by observing the time of eclipses in different localities. Hipparchus 
believed that an extensive series of observations should be carried out in order 
to ascertain, by mathematical and astronomical means alone, latitudes and 
longitudes of a large number of places. To facilitate such a survey he prepared 
tables of lunar eclipses and tables to aid in the determination of latitudes, but 
the practical difficulties of the undertaking were too great and the work was 
never completed. In fact, throughout antiquity the total number of places 
whose position had thus been accurately determined probably does not exceed 
half a dozen, if it is as many. Pliny gives an account of two different occasions 
when observations were made of the same eclipse at two different places. He 
says that at the time of the battle of Arbela the moon was eclipsed at the second 
hour of the night, when at the same hour it was rising in Sicily. He also speaks 
of an eclipse of the sun that was seen in Campania between the seventh and 
eighth hours and in Armenia between the eleventh and twelfth, indicating 
a difference in longitude of four hours, or 60. The actual distance is no more 
than half of this. Ptolemy also cites the eclipse of 331 B.C. as giving the distance 
between Carthage and Arbela. . . . Much greater accuracy was attained by 
the Arabs in their calculations of longitude and some of their figures were passed 
on to the Western world in astronomical tables during the twelfth and thirteenth 
centuries. 

The use of eclipses depends on being able to reckon when the sun, the earth, 
and the moon will be in line with one another as seen at some particular 
place. This means being able to calculate the relative positions of the sun 
and moon as seen from the earth by analysis of recorded observations. An 
analogous method depends on the fact that the moon retreats in its monthly 
course through approximately 360-^-30, or 12 degrees a day. So its position 
with reference to the fixed stars changes appreciably in a couple of hours, 
even as judged by the use of very simple instruments. Thus the moon is a 
clock which registers short intervals of time. Once we have discovered how 
to map out the moon's relation to the apparent rotation of the fixed stars, 
there are two different ways in which the moon's motion may be used to 
compare local time with time at a standard observatory. 

The discovery of how to map the position of the heavenly bodies was 
therefore a necessary preliminary to scientific earth survey. The wealth of 
astronomical knowledge amassed by the Alexandrian astronomers was not 
merely the by-product of idle curiosity, nor a lopsided hypertrophy of 



82 Science for the Citizen 

leisure. In constructing the first maps they were forced to rely on much more 
devious methods than we employ in an age of spring clocks. Getting a fairly 
good estimate of latitude inland in A.D. 1935 in the northern hemisphere is 
a very different thing from finding a latitude at sea on a particular day in the 
year 300 B.C. To begin with, there was no bright star very near to the celestial 
pole at that date. Even today the presence of Polaris 1 from the North Pole 
does not help navigation in the southern hemisphere. The sun is often 
obscured by cloud at noon so that it is highly unsatisfactory to rely on one or 
even two methods for finding one's bearings. The art of navigation, or of 
overland survey for imperialist campaigns, has to make use of all the relevant 
information which any visible celestial object can provide. 

While important contributions to optics., mechanics, and medicine are 
to be credited to the Alexandrian culture, its supreme achievements are all 
related directly or indirectly to the discovery of a scientific basis for earth 
survey. Alexandria was a centre of maritime trade. It was also a cosmopolitan 
product of Greek Imperialism, and thereafter the cultural Mecca of the 
Roman Empire. Knowledge of latitude grew out of the mariner's practice 
of steering his course by the heavenly bodies and noticing the changing 
elevation of the Pole Star in coastal sailing northwards or southwards across 
the Mediterranean or beyond the Pillars of Hercules. Knowledge of longitude, 
as the previous citation suggests, came from the arts of war, and did not 
enter into the practice of navigation till a far later date. Estimates of long 
distances depended on information from imperialist campaigns. Scientific 
geography was in part a by-product of the practice of navigation, in part a 
by-product of imperial expansion; and it was made possible when the 
pre-existing lore of a ceremonial caste became the common possession of 
mankind. Brilliant innovations in mathematics arose in close relation to the 
same group of problems. The trigonometry of Archimedes and Hipparchus, 
the algebra of Theon and Diophantus, can be traced to the inadequacies of 
Platonic geometry and Greek arithmetic as instruments for handling the 
large-scale measurements which Alexandrian geodesy and astronomy 
entailed. 

The principal discoveries which form the basis of the Ptolemaic system 
may be taken under four headings : the measurement of the size of the earth, 
the construction of universal star maps based on the principle of latitude 
and longitude, the introduction of latitude and longitude for terrestrial 
cartography, and the first estimates of the distances of the moon and sun 
from the earth. 

THE SIZE OF THE EARTH 

The first estimate of the earth's circumference was made by Eratosthenes 
(c. 250 B.C.). As librarian of Alexandria he had access to information from 
which he was able to compare the sun's altitude at two situations, one due 
south of the other (Fig. 38). On the summer solstice the noon sun was 
mirrored in a deep well near Syene just below the first cataract of the Nile 
on the tropic of Cancer. The noon sun was therefore at the zenith. On the 
same day at Alexandria, 500 miles due north, the obelisk shadow showed 
that the sun was 7| from the zenith, Since the sun's rays are parallel, this 



Pompey's Pillar 




Zat.3/ 
SOO miles n. 



Equator 




of tlbG. earth's 

ctrcuirtferenc'e, 



\^L^^ ~Nocm sun directly 
1=_ overh&ad atSyew 
m Tropic of Cancer 
~& (zenith 



'Noon sun at 
Alexandria 

June 21^ 
(zenith dist. 




FIG. 38 

Note that at noon the sun lies directly over the observer's meridian of longitude. 
Syene and Alexandria have nearly the same longitude. So the sun, the two places, 
and the earth's centre, may be drawn on the same flat slab of space. 



means that the arc, or as we now say the difference of latitude, between 
Syene and Alexandria is approximately 7^, i.e. 7| 360 = of the 
entire earth's circumference. Hence, the difference between Alexandria and 
Syene (approximately 500 miles) is of the circumference of the globe. So 

the earth's circumference is about 48 x 500 = 24,000 miles. The distance 
was given in stadia (the length of the games stadium), and was probably 



8 4 



Science for the Citizen 



based on the average day's journey of an army on the march. The result 
agrees within 4 per cent with modern determinations. 

In his book, The World Machine^ Carl Snyder quotes some other figures 
relative to early estimates of the earth's size : 

There was a curious tradition, preserved by Achilles Tatius, that the 
Chaldeans had measured the earth in terms of a day's march. They said if a 
man were able to walk steadily, and at a good pace, he would encompass the 
earth in one year. They counted that he would do 30 stadia (about 3 miles) 
an hour, and so computed the great circle of the globe at 263,000 stadia, which 
was very close to the estimate made by Eratosthenes. . . . Eratosthenes' 




_ rays ^ 
Irani Canopus* 



FIG. 39 

How Poseidonius measured the circumference of the earth. The difference between 
the zenith distances of the same star when measured at its meridian transit from two 
stations is the difference of latitude between the two stations. If one is due north 
of the other this is the number of degrees in the arc of the earth's circumference between 
the two stations. Since Canopus grazed the horizon (Z.D.=90) at Rhodes the differ- 
ence in latitude between Alexandria and Rhodes was 90 (Z.D. at Alexandria). 

figure was 250,000 (or 252,000); Hipparchus wished to increase this to 275,000; 
Poseidonius to reduce it, fixing it at 240,000 stadia according to Cleomedes, at 
180,000 according to Strabo. This last figure was that adopted by Ptolemy, and 
this and other errors of Ptolemy were the basis of Columbus* belief that India 
was near. Had he known the true distance, possibly he never would have 
sailed. ... Is it the impression that we have here merely an intellectual 
conception that the meaning of it in no wise came home to any of the ancients 
that there was no vivid sense of new continents, new worlds to explore? 
Listen, then, to a passage of old Strabo; he is telling of the ideas of Eratosthenes, 
who, he says, held "that if the extent of the Atlantic Ocean were not an obstacle, 
we might easily pass by sea from Iberia (i.e. Spain) to India, still keeping in 
the same parallel (of the temperate zone), the remaining portion of which 



Pompey's Pillar 



parallel, measured as above in stadia (252,000) occupies more than a third of 
the whole circle; since the parallel drawn through Athens, in which we have 
taken the distances from India to Iberia, does not contain in the whole 200,000 
stadia." 

To get his measured distance Poseidonius relied on mariners' estimates 
of a straight course northwards across the Mediterranean. Like that of 
Eratosthenes, his method depends on the fact that the difference of latitude 
between two places is equivalent to the difference of zenith distance or 
altitude of the same heavenly body when it crosses the meridian (Fig. 39) 5 



Pole 




FIG. 40. SMALL CIRCLES OF LATITUDE ON THE TERRESTRIAL AND OF DECLINATION 

ON THE CELESTIAL SPHERE 

Just as latitude is the angle of elevation north of the equator of a point on the earth's 
surface, so declination is the angle of elevation north of the celestial equator of a 
point on the celestial sphere. Latitudes and declinations south of the equatorial plane 
are thus reckoned as negative. 



Poseidonius measured the zenith distance of the star Canopus at Alexandria 
and on the island of Rhodes, situated roughly 350 miles north of the former. 
Canopus, after Sirius the brightest star in the sky, just grazes the horizon 
at Rhodes, and is invisible further north. 

According to the best measurements, the earth's diameter is 7,900 miles 
along the polar axis and 7,926-7 miles across the equator. The geometric 
mean of the two is 7,913-3 miles. The flattening at the poles about which 
we heard so much at school amounts to a difference of less than fifty miles 
between the two axes. Thus the circumference along the Greenwich meridian 
is 77 X 7,913-3 = 24,860 miles, and along the equator, -n x 7,926-7 = 24,902 
miles. Once the size of the earth is known, it is an easy matter to calculate 



86 



Science for the Citizen 



the distance which separates any two places, if their latitude and longitude 
are also available. The position of a ship at sea with reference to any port 
can be calculated within a few yards. 




FIG. 41 A STAR AT MERIDIAN TRANSIT LIES IN THE SAME PLANE AS THE OBSERVER, 
THE EARTH'S POLES, THE ZENITH, THE SOUTH AND NORTH POINTS OF THE HORIZON, 

AND THE EARTH'S CENTRE 

A straight line which docs not lie on some particular flat slab of space (or plane) can 
only cut it once. A straight line which passes through more than one point on a plane 
therefore lies on the same plane as the points through which it passes. The plane 
bounded by the observer's great circle of longitude and the earth's axis includes as 
points the earth's centre, observer, and the earth's poles. The line joining zenith and 
observer passes also through the earth's centre, i.e. through more than one point on 
this plane. Hence it lies wholly on the plane. The north and south points of the 
horizon lie in the same plane as the meridian of longitude through the observer. 
So south and north points, zenith, and observer, are all in the same plane with the 
earth's centre and poles. A circle can only cut a plane in which it does not itself lie at 
two points. The circle drawn through the north and south points and the zenith passes 
through more than two points in the same plane, and therefore lies wholly on it. So 
any point on this imaginary circle (the celestial meridian) is also on it. 

THE CONSTRUCTION OF A STAR MAP 

Today we are all brought up to use maps. An acquaintance with latitude 
and longitude is part of the equipment of a citizen for everyday life in an 
age of widespread communications over large distances. We shall therefore 
find it more easy to envisage the construction of a star map from what we 



Pompey's Pillar 87 

have already learned in using an earth map. Imagine a long stick connecting 
the celestial sphere with the centre of the earth. If we rotate it (Fig. 48) in 
a great semicircle from pole to pole, it will trace out a meridian of longitude 
on the surface of the earth, and a corresponding semicircle which is called 
a meridian bright ascension on the celestial sphere. All observers on the same 
semicircle of longitude will have the same local time. They will all see any 
celestial body of the same right ascension crossing the meridian at the same 
moment. The difference between the longitudes of any two observers will be 
the hour angle difference between the corresponding semicircles of right 
ascension coinciding with their celestial meridians at the same moment. If 
we imagine the stick to rotate (Fig. 40) so that it always cuts the earth's surface 



Star decluia^Lcm d. 




Centre 
06 EartK 

FIG. 42 

Two stars in the northern half of the celestial sphere, one crossing the meridian 
north, the other south of the zenith. If the star crosses north: Declin. = Lat. of 
observer + meridian zenith distance. If it crosses south: Lat. of observer = Decli- 
nation + meridian zenith distance, i.e. Declin. Lat. of observer meridian 
zenith distance. The first formula holds for all cases if we reckon as negative zenith 
distances south of the zenith, and latitudes or declinations south of the equator. 

at the same distance from the equator, it will trace out a circle of latitude on 
the earth's surface, and a corresponding circle of declination on the celestial 
sphere. As the celestial sphere rotates around its axis in the plane of the 
equator, any heavenly body on this circle of declination will be seen exactly 
at the zenith when it crosses the meridian of any observer on the corresponding 
circle of terrestrial latitude. A star on any other declination circle will have 
the same meridian altitude for all observers on any one circle of latitude. 

If the declinations and right ascensions of the stars are tabulated, we can 
use any one of them to find latitude (Figs. 42 and 43) or longitude. The 
practical advantage of this is that : (i) if we are travelling rapidly, we can 
check our position at frequent intervals, (ii) we are less dependent on continu- 
ously fine weather, (iii) we can make the necessary observations at any time 
between sunset and sunrise since there is always some star near the meridian. 
The construction of a rough star map which will give latitude and longitude 
within a degree is comparatively simple. 



88 



Science for the Citizen 



To get the declination of a star in the northern hemisphere, find its altitude 
(measured from the north point) when it crosses the meridian. This will be 
its maximum altitude taken at intervals of a minute round about the time 
when its azimuth is nearly zero. Subtract the local altitude of the Pole Star. 

FIG. 43. LATITUDE, DECLINATION, AND ZENITH DISTANCE, AT MERIDIAN TRANSIT 

In both parts of the following figure, COP = ZON, since COP is the right angle 
between the celestial equator and the polar axis, and ZON is the right angle between 
the zenith and the horizon. 




S O 

Star A transits north of ii 12 
For a star (A) which transits north of the zenith : 

COP - AOP = ZON - AOP 
i.e. Declin. z.d. -j- Lat. 
or Lat. Declin. z.d. 




3 tar B transits south of t/ic 'Zenith 
For a star (B) which transits south of the zenith : 

COP - ZOP - ZON - ZOP 
i.e. Declin. + z.d. Lat. 
or Declin. = Lat. z.d. 

The first two formulae hold for both cases if zenith distances south of the zenith are 
reckoned as negative. 

This gives us its angular distance from the Pole Star, i.e. its north polar 
distance within 1. To the same order of precision the declination is 
90 N.P.D. (Fig. 24). The error is due to the fact that the pole is not 
exactly located at the point round which the stars appear to revolve. Polaris 
itself describes a tiny circle of 1 round the true celestial pole. The mean of 



Pompey's Pillar 89 

two determinations of the altitude of the Pole Star taken with an interval 
of six months at the same hour of the night gives a correct value. In principle 
this is what we should have to do, if we wanted to locate the pole of the 
southern hemisphere, because there is no bright star near it. There was 
no bright star very near the north celestial pole in the time of Hipparchus. 
The altitude of the celestial pole (P), whether south or north (Fig. 44), is 
simply the average of the altitude (A) of any circumpolar star at its upper 
culmination and the altitude (a) of the same star at its lower culmination, i.e. 

P = ?>(A + a) 



^ Star at Upper 
J\ CuLniLtLcdioji 



Star at icnver 




Horizon Plane 



Lat. = 

FIG. 44 

The celestial pole is half-way between the positions of a circumpolar star at its upper 
and lower culmination. Hence the angle (B) between the celestial pole and a circum- 
polar star is twice the angle between its positions at lower and upper transit. So if the 
altitude of the star at lower transit is a 3 and at upper transit A, A a + 2B. The 
observer's latitude is the altitude of the celestial pole. 

Lat. = a + B = a + i (2B) 

- a + J (A - a) = i (A -f a) 

Having got P, the south polar distance of any star whose meridian altitude 
is M, is M P, and its south declination is 

90 - S.P.D. 

It will help you to visualize the meaning of declination if you see how 
the declination of a heavenly body affects its direction of rising, setting, 
transit, etc., by considering the appearance of the heavens at some particular 
latitude, e.g. Lat. 50 N. in Figs. 45 and 46. You see for instance that the 
plane of the midsummer sun's (declin. = + 23|) apparent diurnal path cuts 
the horizon plane north of the midpoint between the north and south extremi- 
ties of the horizon, so that the sun rises and sets north of the east and west points 
between March 21st and September 23rd and transits on June 21st 26J south 
of the zenith. Between September 23rd and March 21st the declination is nega- 
tive, being 23J on December 21st, and the sun rises and sets towards the 
south. It transits only 16J above the horizon at Lat. 50 N. on December 21st. 

Without making any measurements at all, you can get a good estimate of 
your latitude (L) if you know the stars by name and have an almanac at hand. 
The following simple deductions illustrated in Fig. 46 apply to latitudes north of 



90 Science for the Citizen 

45 N. A corresponding figure will show you how to modify rules (a), (b) and 
(c) for latitudes farther south. 

(a) If the declination of a star is greater than -i L, the star is circum- 
polar and transits north of the zenith. 

(b) If the declination of a star is exactly -f L, it is circumpolar and transits 
at the zenith. 

(c) If its declination lies between -\- L and (90 L), it is circumpolar 
and transits south. If its declination is exactly (90 L), it just grazes the 
northern horizon at lower transit. 



"ft 




FIG. 45 

The sun at latitude 50 N transits on June 21st 26 S from the zenith and 03 T above 
the southern horizon. On December 21st it transits only KU above the southern 
horizon. 

(d} If its decimation lies between (90 L) and ', it rises and sets north 
of the east and west points of the horizon, and transits south. 

(e) If its declination is exactly 0, it rises and sets due east and west and 
transits south. 

(/) If its declination lies between and (90 L), it rises, sets, and 
transits towards the south. 

(g) If its declination is exactly (90 - L), it just grazes the southern 
horizon at upper transit. 

It follows from these considerations that a table of declinations fixes your 
north latitude, if you can distinguish any star which transits at the zenith, any 
star which just grazes the northern horizon at lower transit, or any star which 
just grazes the southern horizon. In a similar way, you can find your south 
latitude if you live in the southern hemisphere. The previous figure (Fig. 45) 
shows you that the lengths of day and night are equal on the equinoxes, and 
that the lengths of night and day are reciprocally equivalent on alternate solstices. 
To get a picture of the rising and setting of a star at different times of the 
year, you need to know its R.A. and that of the sun, From the diagram you 



Pompey*s Pillar 



will see that if a star has south declination, it will never, at the latitude specified, 
be above the horizon for half its diurnal course, i.e. twelve hours. This means 
that if it is a winter star, it will traverse its complete course over the horizon in 
less time than the duration of darkness which is then greater than twelve hours. 
To lay off the other coordinate of a star corresponding to longitude, we 
have to choose a base line in the heavens like the Greenwich meridian, which 
is usually taken as for terrestrial position. The one chosen is the semi- 




Star (a) 
Star 
Star (c) 



Star 0) 
Star (/) 



FIG. 40 
At Latitude 50 N. 

declination greater than .30,, is circumpolar, and transits north of the zenith. 

declination exactly SO , transits at the zenith and is also circumpolar. 

declination 40 (N.P.D.^ 50)., transits south of the zenith at its upper cul- 
mination and just grazes the horizon at its lower. 
Star (d} declination less than 40, greater than 0, transits south, rises in the north- 
east and sets in the north-west. 

declination exactly 0, rises due east and sets due west, transits south. 

declination less than 0% greater than 40, rises in the south-east, sets in 

the south-west and transits south. 
Star (g) declination exactly ~~ 40% just grazes the horizon at transit. 

circle of right ascension on which the sun lies at the exact moment when it 
crosses the equator on the vernal equinox (Figs. 47 and 48). Where the 
equinoctial intersects with the ecliptic in the spring position of the sun is 
called the First Point of Aries. This is the celestial Greenwich. The Right 
Ascension of a star is the hour angle which separates it from the Right Ascen- 
sion circle which passes through the First Point of Aries. It is measured 
eastwards. So it is obtained by subtracting the time* at which the First Point 
of Aries crosses the meridian from the time at which the star under observa- 
tion crosses the meridian. It is therefore usually given in hours rather than 

* Strictly sidereal (see p. 60). The error in counting it as solar for short periods is 
neglected. 



92 Science for the Citizen 

degrees (15 = 1 hour, or 1 = 4 minutes). This practice may at first confuse 
you, because degrees, like hours, are divided into minutes and seconds. So it is 
always important when speaking of seconds or minutes to make sure whether 
units of time or angular measure are being used. The signs ' and " are used 
for angular minutes and seconds. 

For rough purposes the method of finding the R.A. of a star is therefore 
as follows. Determine the time at which any star crosses the meridian after 




FIG. 47 

Noon at Greenwich on March 21st. Showing relation of R.A., longitude, and time. 
At noon the R.A. meridian of the sun in the celestial sphere is in the same plane as 
the longitude meridian of the observer. If you are 30 W. of Greenwich the earth must 
rotate through 30 or /j of a revolution, taking 2 hours, before your meridian is in 
the plane of the sun's, or the sun must appear to travel through 30 before its meridian 
is in the same plane as yours. Hence your noon will be 2 hours behind Greenwich. 
A clock set by Greenwich time will record 2 p.m. when the sun crosses your meridian, 
i.e. at noon local time. If the date is March 21st when the sun's R.A. is zero a star of 
right ascension 6 hours will cross the meridian at 6 p.m. local time. If it crossed at 
8p.m. Greenwich time your clock would be 2 hours slow by Greenwich, so your 
longitude would be 30 W. The figure shows the anti-clockwise rotation of the stars 
looking northwards, so the south pole is nearest to you. 

sunset on the vernal equinox. The sun is approximately at the First Point 
of Aries at noon on that day. So the number of hours which elapse between 
noon and the time when the star crosses the meridian after sunset is approxi- 
mately its right ascension. If a star is not visible on the night following noon 
on the vernal equinox, we have only to compare its time of transit at a 
convenient season with that of one which is visible throughout the year, e.g. 
a in Ursa Major. If it transits before the standard star, we subtract the 
difference from the R.A. of the latter. If it transits later, we add the difference. 
If it rains or the sky is overcast on the night following the vernal equinox, we 
can use the sun as our standard star, taking its right ascension as 6 hours on 
June 21st, 12 hours on September 23rd, or 18 hours on December 21st, 
and so forth. Since the sun retreats eastwards through 360 in 365 days, its 



Pompey's Pillar 



93 



R.A. increases from at the vernal equinox to 360 at the next by roughly 1 
per day. 

Hence, if the R.A. of Betelgeuse is stated to be approximately 5 hours 
50 minutes, this means that Betelgeuse transits 5 hours 50 minutes after the 
First Point of Aries (T), or approximately at 5.50 p.m. on the day following 
the vernal equinox (Fig. 48). It will therefore be invisible at the time of 
transit since the sun has not quite set. A month earlier (30 days) the sun 
would not yet have reached T in its annual retreat eastward in the ecliptic., 
and its R.A. would be 360 30 = 330 approximately, or in time units 
22 hours. Thus the sun would transit 2 hours before T, and the star would 



.oi^-'^X.V 

ip&'ff 



W &/ <2 
V ^/ ^7 
Y cf #/ 
/ V *?/ 

/ t 7 

/ / ^Iurn4Ll2^^, 

/ MB^fe^ 





/ 



FIG. 48 

The star shown (R.A. 6 hours) makes its transit above the meridian at noon on June 2 1st, 
and midnight on December 21st> i.e. it is a winter star like Betelgeuse. 

therefore cross the meridian 7 hours 50 minutes after the sun, i.e. at 7.50 p.m. 
A star will transit at midnight when its R.A. differs from that of the sun by 
180, or 12 hours. Hence Betelgeuse would transit at midnight when the sun's 
R.A. is 17 hours 50 minutes, i.e. when the sun's R.A. is about 10 minutes 
behind its R.A. on December 21st. Since 10 minutes in time represents 
(10 -f- 60) x 15 = 2|, this is approximately 2 days before December 21st, 
i.e. about December 19th. 

Alternatively you can look on the R.A. of a star as a way of determining 
local time at night. Thus, if the R.A. of Betelgeuse is given as 5 hours 
50 minutes, and the date is November 1st, we can set a watch which has 
stopped by finding the moment at which it crosses the meridian. On Novem- 
ber 1st, 39 days after the autumnal equinox (when the sun's R.A. is 12 hours 



or 



94 Science for the Citizen 

180) the sun's R.A. is approximately 219, or ( 12+ ] hours=]4 hours 

V 15 y 

36 minutes. The sun therefore transits 24 hours 14 hours 36 minutes = 9 
hours 24 minutes before T, or 15 hours 14 minutes before the star. When 
Betelgeuse transits the local time is therefore 15.14 p.m., or 3.14 a.m. You will 
see from this that a single glance at a table of R.A. tells you whether a star 



T (First point of Aries' 





FIG. 49. STAR MAP (OR PLANISPHERE) TO ILLUSTRATE RELATION OF RIGHT ASCENSION 

TO LOCAL TIME OF TRANSIT 

If the sun's R.A. is x it transits x hours after the zero R.A. circle (though the First 
Point of Aries). If the star's R.A. is y, it transits y hours after r . The star therefore 
transits y x hours after the sun,, i.e. its local time of transit is (y - - x). Hence 

Star's R.A. Sun's R.A. - Local time of transit 

It may happen that the difference is negative, as in the example in the figure, the 
local time of transit being 15 hours 9 minutes, i.e. 15 hours 9 minutes before noon 
which is the same as 8 hours 51 minutes after noon (8.51 p.m.). The figure shows that 
the sun transits 3 hours before T, and the star 5 hours 51 minutes after, making the 
time of transit as stated. The orientation is the same as in Fig. 47. 

is in the ascendant in summer, winter, spring, or autumn. That the R.A. 
of Betelgeuse is 5 hours 50 minutes means that Betelgeuse occupies the same 
position in the celestial sphere as the sun does approximately 10 ~ 4, or 
2J- days before June 2lst (when its R.A. is 6 hours), i.e. about June 19th. 
Hence it will transit months later at midnight (about December 19th), 
and is a winter star. 
In taking the position of T as the sun's position at noon on the vernal 



Pompey's Pillar 



95 



equinox there is a small inaccuracy.* The way in which we actually determine 
the vernal equinox does not mean that the sun is exactly over the equator 
at local noon on that day. Hence the sun is not exactly at the First Point of 
Aries when it crosses the meridian. The vernal equinox is chosen as the day 
on which the sun at noon is nearer to the plane of the equator than it is on 
the day before or the day after. It is exactly at the First Point of Aries at 
some time on the vernal equinox, that is to say., within 12 hours before noon 
or after noon. Since it slips back eastwards at the rate of 1 per day of 24 
hours, the method given does not involve an error of more than | or 2 
minutes of time in right ascension. The sun's right ascension at noon on 
the vernal equinox, therefore, lies between 23 hours 58 minutes and 
hour 2 minutes. With home-made instruments we can be very well satis - 




ZENITH 



FIG. 50 

The sun crossing the equator. Above just before March 21st, sun south of the equator, 
declination (noon z.d. of sun as seen by an observer at the equator) negative. Below 
just after March 21st declination positive. 

fied with an error as small as this. It only involves an error of -J in 
longitude. At the equator that means a distance of roughly (2rr x 4,000) -: 
(2 x 360), or about 35 miles in a measurement round the earth. 

For more accurate determination of the R.A. of a star we require to know 
when the sun "crosses the equator," i.e. when it is exactly at the point when 
the ecliptic intersects with the equinoctial. Over the equator the sun is highest 
at noon on the equinox, being south of the zenith (z.d. negative) on the day 
preceding the vernal equinox and north (z.d. positive) on the day following it 
(Fig. 50). Since the declination of a heavenly body is its z.d. in meridian transit 
at the equator, the declination changes from negative to positive through 
zero. The time at which the sun crosses the equator can be found approximately 
thus. Suppose the declination of the sun by z.d. at noon is known for the day 
before and after the equinox, so that : 

Decimation at noon March 20th = D 
Declination at noon March 22nd = -f d 

* See also footnote on p. 91. 



96 Science for the Citizen 

In 48 hours the increase in declination is therefore : 

d - (- D) - d + D 
In t hours it is : 

d -^T Kt 

If t hours after noon on March 20th is the time when the sun crosses the 
equator, the declination increases from D to in t hours, i.e. the increase 
isD. 



__ 48D 

t ~ r" 



If the time is reckoned from noon on March 21st, it is 

24(D - d) 



t - 24 = 



D 



Suppose, then, that the sun crosses x hours after noon. In these x (= t 24) 
hours it advances east towards T at the rate of 1 or an hour angle of 4 minutes 

A y v 

per day, i.e. it will be -~ = - minutes in time units west of T at noon. Therefore 

x 
p crosses the meridian - minutes after noon, and the sun's R.A. at noon is 

x 6 
minutes expressed in time units. This quantity will have to be added to 

6 

the difference in sidereal time after noon when a star crosses the meridian to 
get its true R.A. So if the time of transit of a star is T hours after noon on the 

equinox, its R.A. will be: T + ( ^) = T - ~. According to Whitaker's 

O t) 

Almanack, 1934,, the declinations of the sun at noon were: 

March 20th 19-2 minutes (angular units) 
March 22nd -f 28-2 minutes (angular units) 

Hence the time after noon March 21st when the sun crossed was: 
24(D - d) -9 x 24 



D -\ d 47-4 



= 4-5 approx. 



i.e. the sun crossed the equator 4| hours before noon or at 7.30 a.m. on 
March 21st. Hence the time at which T crosses the meridian is 

4-5 

- minutes = 45 seconds (approx.) 

D 

before noon and the R.A. of the sun at noon is 45 seconds. This must be added 
to the time which elapses between noon and the next transit of a star to get its 
correct R.A. in time units. 

It is important to bear in mind that mapping stars in this way is merely 
a way of telling us in what direction we have to look or to point a telescope 
in order to see them. The position of a star as shown on the star map -has 



Pompey's Pillar 97 

nothing to do with how far it is away from us. If you dug straight down 
following the plumbline, you would eventually reach the centre of the earth; 
and the bottom of a straight well, as viewed from the centre of the earth, 
if that were possible, would therefore be exactly in line with the top. It has 
the same latitude and longitude as the latter, though it is not so far away 
from the earth's centre. The latitude or longitude of the bottom of a mine 
is the latitude and longitude of the spot where the line joining it with the 
centre of the earth, if continued upwards, cuts the earth's surface. So the 
declination and right ascension of a star measure the place where the line 
joining the earth's centre and the star cuts an imaginary globe whose radius 
extends to the farthermost stars. In a total eclipse the sun and the moon 
have the same declination and R.A. just as the top and bottom of a mine 
have the same latitude and longitude. This means that the sun and moon 
are directly in line with the centre of the earth like the top and the bottom 
of a mine. If we are only concerned with the direction along which we have 
to look or tilt our telescope to see a star, we may therefore treat stars as if 
they were all placed on the surface of one and the same celestial sphere. 



INTERPRETATION OF THE STAR MAP 

Having determined the relative positions of a sufficient number of bright 
stars Hipparchus tabulated 1,080 fixed stars in 150 B.C. we can construct 
a star map like that of the northern celestial hemisphere in Fig. 51. Such a 
map or planisphere embodies in a compact form all the requisite data, set 
forth with more precision in nautical almanacs, for finding the position of 
a ship at sea.* 

To find latitude from the declination of any star which is near the meridian 
at the time when we require to know where we are, we make use of the 
simple rule explained in Fig. 43, i.e. 

Declination = Latitude + Z.D. or Latitude = Declination -~ Z.D. 

To make this apply to all circumstances, it is necessary to reckon z.d. with 
opposite signs north and south of the zenith, and latitude or decimation with 
opposite signs north and south of the equator. By agreement, north measure- 
ments are reckoned positive, south measurements negative. Thus, as you will 
see from Fig. 43, with northern latitudes we may distinguish three cases: 

(a) Star north of the equator and zenith: Decimation and z.d. both 
positive : L D Z 

(6) Star north of equator, south of the zenith: declination positive, z.d. 
negative: L = D -f- ( Z), .". L = D Z 

(c) Star south of the equator and zenith : declination and z.d. both negative : 
L = ( Z) ( D), /. L = D - Z 

For latitudes south of the equator the three corresponding cases are : 
(a) Star north of the equator : declination and z.d. both positive : 
(- L) - Z - D, /. L = D - Z 

* The first hint of such a model seems to have been given by the Athenian Eudoxus. 

D 



98 Science for the Citizen 

(fc) Star south of the equator, north of the zenith: declination negative 
and z.d. positive: ( L) = ( D) + Z, /. L = D Z 

(c) Star south of equator and zenith : declination and z.d. both negative : 

/ T \ / T-\N / 7\ . T T\ *J 

( L.) = ( D) ( ), . . JL = U Z 

If we know the declination circle of any star on the star map, we have 
only to determine its zenith distance at meridian transit to obtain our lati- 



TVlarc/i 21 
r 



U 



' Aldebaran (UL Taurus) 
' 




Sept. 25 

FIG. 51 

Star map or planisphere showing the position of a few of the principal stars and 
constellations in the northern celestial hemisphere in circles of declination and radii 
of right ascension. The sun's track in the ecliptic and its position at the solstices and 
equinoxes are also shown. 

tude. For example, on a certain night the altitude of the star a in Cassiopeia 
from the northern horizon was found to be 83 10' at its upper culmination. 
The declination of a Cassiopeiae is given in the Nautical Almanac as 56 11 '. 
From the observed meridian altitude the zenith distance of the star at the 
ship's latitude is found to be 90 83 10' = + 6 50'. Since 

Latitude = Declination Z.D. 
the latitude of the ship was 

56 11'- 650'=r 4921'N. 



Pompey's Pillar 99 

Now that we have clocks, the determination of longitude may be illustrated 
by the following example in which the data given are not very accurate. 
On January 21st the star Betelgeuse is seen crossing the meridian when 
the ship's chronometer (Greenwich Mean Time) registers 11.30 p.m. The 
R.A. of Betelgeuse is 5 hours 52 minutes (to the nearest minute). Hence 
Betelgeuse crosses the meridian 5 hours 52 minutes after T . If we are without 
an almanac we can make a rough estimate of our longitude as follows. On 
January 21st the sun has moved through i%th of its apparent annual retreat 
since December 21st, when its R.A. is 18 hours. So the R.A. of the sun on 
January 21st is approximately 20 hours; and T crosses the meridian 4 hours 
after the sun, i.e. at 4 p.m. Hence Betelgeuse crosses the meridian at 
5 hours 52 minutes + 4 hours = 9 hours 52 minutes after local noon. So 
by local time its time of transit is 9.52 p.m. Neglecting the "equation of 
time," local time is therefore 1 hour 38 minutes or If hours behind Green- 
wich, and the ship is approximately ] X 15 = 25 W. of Greenwich. 

The almanac for 1937 tells us that the sun's noon R.A. at Greenwich on 
January 21st is 20 hours 13 minutes. So its R.A. at 11.30 p.m. is about 
2 hours 15 minutes; and the local time at which Betelgeuse transits is 3 hours 
45 minutes + 5 hours 52 minutes, i.e. 9 hours 37 minutes. The almanac 
also states that we must add 11 minutes 24 seconds ("equation of time") to 
"apparent" (i.e. sundial) time to get mean time. We must therefore take away 
11 minutes 24 seconds from the chronometer time to get true solar local 
time at Greenwich. Hence the Greenwich solar time of transit is 1 1 hours 
19 minutes. Thus the ship's time is 11 hours 19 minutes 9 hours 37 minutes 
= 1 hour 42 minutes slow. Counting 4 minutes as equivalent to one degree a 

102 
more accurate estimate of the longitude is therefore = 25 J W. 

From these examples you will see that with modern methods of recording 
time, a ship's position can be determined at noon, sunset, midnight, and 
sunrise, or at any hour between sunset and sunrise, if the weather is fine. 
When the sky is overcast so that the familiar constellations are difficult to 
recognize, the star map also helps us to locate a particular star in a favourable 
position for observation, provided we already know our approximate bearings 
by a recent determination. In navigation the exact moment of transit of a 
star is not easy to record. It is usual to note the times when the star has the 
same altitude before and after transit, i.e. east and west of the meridian. The 
time of transit is midway between. 

Looking at the map in Fig. 51 you will see that the summer star Arcturus 
of Browning's poem lies near the right ascension line XIV, i.e. it comes 
on the meridian about 2 a.m. on March 21st, about midnight April 21st, 
about 8 p.m. (or about sunset) on June 21st. On October 28th it is on the 
meridian about noon. So it will not be visible high in the heavens at any 
time on the latter date. To get a picture of its hours of rising and setting, 
we can make a rough construction by taking into account its declination 
circle (approximately 20). The diagram (Fig. 52) shows that the sun rises 
and sets about 7 a.m. and 5 p.m. respectively. Arcturus rises and sets at 
4 a.m. and 8 p.m. respectively. So Arcturus will be visible in the eastern 
sky for three hours before sunrise and in the western sky after sunset. 



ioo Science for the Citizen 

The use of the star map to determine the azimuths of rising and setting, 
as also the times of rising and setting at different seasons, will be set forth 
more fully in Chapter IV, p. 196. The method of finding the moon's R.A. 
and declination on days of the month, when it does not transit after dark, 
will also be explained in the same context. What has been said about Arcturus 
suffices to illustrate the most characteristic features of the lunar cycle. The 
moon's R.A. increases by 360 degrees in a period a little less (Fig. 5.3) than an 
ordinary month reckoned from new moon to new moon. It has to move through 
a little more than 360 to catch up with the sun, because the sun's R.A. is 
slowly increasing at about one-twelfth the rate at which the moon recedes 
eastwards among the stars. If the moon's retreat eastwards occurred in the plane 
of the celestial equator, it would rise due east at noon, transit at C p.m. and set 



I- - -\ -\ .% Oct. 2^ 




FIG. 52 

Construction to show how the times of rising and setting of a star (Arcturus) can be 
deduced from its declination (20 N.) at a particular latitude (50 N.) on a particular 
day of the year (October 28th). 

due west at midnight on the first quarter day; rise due east at 6 p.m., transit at 
midnight and set due west at 6 a.m. when full j rise due east at midnight, transit 
at 6 a.m. and set due west at noon on the last quarter day. The time and direction 
of rising and setting in the various phases of the moon vary from month to 
month because the moon retreats in an oblique path, like that of the sun, 
among the zodiacal constellations. As will be explained below (see also Fig. 53), 
the trace of the moon's cycle on the star map lies very close to the ecliptic. So 
the moon's cycle exhibits certain analogies with that of the sun. 

In a northern latitude a star with north declination like Arcturus traces a 
wider arc above the horizon than a star with south declination like Sirius, the 
interval between rising and setting being more than 12 hours for a northerly 
and less than 12 hours for a southerly star. Since the sun remains longer above 
the horizon when it lies in the same direction as the northerly constellations of 
the Zodiac during the summer months, summer days are longer than those of 



Pompey's Pillar 101 

the winter. When the sun is in Aries or Libra (using the terms in their historic 
sense, see p. 64), its declination is changing very rapidly. When in Cancer or 
Capricorn it is apparently retreating in a direction approximately parallel to 
the equator, and hence its declination is changing less rapidly. For this reason 
the length of day and night changes very little about the time of the solstices 
and more noticeably during spring and autumn, when the days are said to be 
"drawing out" or "drawing in." 

Since the moon takes longer to get back to the same position relative to the 
sun (difference between sun's R.A. and moon's R.A. 360) than to regain its 
former position among the fixed stars (moon's R.A. increased by 360) the 
moon's station in the zodiacal belt at any particular phase (position relative 
to the sun) is not the same in two successive months, and since its northerly 
declination is increasing most rapidly when it is in Aries, the "moon day" 
is drawing out, i.e. the duration of its passage above the horizon is increasing 
most rapidly at this stage in its eastward retreat. This means that the time 
of rising on the succeeding night is not so much later as it would be if the 
moon retreated in a plane parallel to that of the equator. Conversely the in- 
terval between sunset and moonrise on successive nights increases rather 
quickly when the moon is in Libra, since the moon's declination is then de- 
creasing most rapidly. Since full moon occurs when the moon's RA. differs 
from that of the sun by 180, the full moon is in Aries when the sun is in 
Libra, i.e. at the autumn equinox. About the full moon ("harvest moon") near 
the autumn equinox the time of moonrise changes very little on successive 
nights. So the interval between sunset and moonrise just after full moon 
changes very little, and there is a relatively long period during which the moon 
is high in the heavens for the greater part of the night. 

Careful observation of the lunar cycle was a task of great social importance 
while calendrical practice adhered to the chimerical attempt to square the 
primitive lunar calendar of a hunting and food-gathering stage in social 
evolution with the stellar or solar calendar of a settled agrarian economy. 
The maritime Greeks (see p. 65) started off with a far more primitive 
calendar than that of the Egyptians; and this fact may have contributed to 
the fruitful fusion of navigational astronomy with calendrical practice. At a 
later date lunar tables had also to be composed for calculation of longitude. It 
will help you to understand how they are made, and, later on, to calculate the 
position of the planets, if you plot the moon's apparent track from one of 
the ephemerides sold for astrological amusement, or from Whitaker's 
Almanack^ as in Fig. 53. This gives its position on the star map from 
January 3rd to 30th in the year 1894. New moon occurs when the moon's 
R.A. is the same as the sun's, full moon when it differs from that of the sun 
by 180. We find from Whitaker that the moon and the sun have the same 
R.A. at some time between noon and midnight on January 7th of the month 
plotted in the figure. The following figures are for midnight, the values 
given in time units being turned into degrees, and obtained by taking the 
average for successive noons. 

Sun's R.A. (S) Moon's R.A. (M) Difference 
(midnight) (midnight") (S M) 

January 6th 288 \ 2871 + 1 

January 7th 289f 300J -11 



IO2 



Science for the Citizen 



The sun's R.A. exceeded that of the moon by roughly 1 at midnight on 
the 6th. By midnight on the 7th the moon's R.A. exceeded the sun's by 
roughly 11. Hence new moon occurred in the early hours of the morning 

"First point 
of Anes op 




FIG. 53 

The moon's course as represented in the star map in the course of a sidereal month 
(see text). The nodes occur at positions occupied by the moon late on January 14th 
and January 25th. 

on January 7th. Full moon occurred soon after noon on January 21st. Thus 
using midnight values as before, Whitaker gives : 

Sun's R.A. Moon's R.A. Difference 

January 20th 303J 114| +188f 

January 21st 304J 130f +173| 

The next new moon occurred in the evening of February 5th. Thus : 

Sun's R.A. Moon's R.A. Difference 

February 4th 319 j 309| +9|- 

February5th 320 j 321f LJ 



Pompey's Pillar 103 

This shows that the period which separates one new moon from the next 
is between 29 and 30 days. The average of many determinations shows that 
it is almost exactly 29| days. 

The interval between two new moons when the sun, moon, and earth 
are in the same line, is called the ordinary or synodic month. The synodic 
month is not the same as the time taken by the moon to revolve in its orbit. 
This time the sidereal month is the time in which the moon gets back to 
the same position with reference to the fixed stars, i.e. the time taken for its 
R.A. to increase by 360, so that it has the same value. The synodic month is 
longer than the sidereal because the sun's R.A. increases in the course of 
the month. When the moon's R.A. is the same as what it was at the last new 
moon, the sun's R.A. is greater. Hence, the moon has to move farther to 
catch it up. The figure shows you that the sidereal month is just over 27 
days. Daily record of its R.A. shows that values recur at average intervals 
of almost exactly 27 J days. The exact time at which the moon regains the 
same R.A. can be easily interpolated from daily records of its transit; and 
since we know the sun's daily change in R.A., we can easily deduce the exact 
time of the new moon. The exact time of recurrence of two new moons, 
i.e. the length of the synodic month, is connected with the length of the 
sidereal month by a very simple formula. The sun moves through 360 of 
R.A. in Y days (one year}. The moon moves through 360 of R.A. in S days 
(one sidereal month) and gains 360 on the sun, so that they both have the 
same R.A. in M days (one synodic month). Thus, 

360 
in one day the sun's R.A. increases -r= 



360 
in one day the moon's R.A. increases ^-~ 

o 

360 360 
in one day the moor's R.A. gains on the sun's by -~ 

360 
But in one day the moon gains in R.A. - 

M 

360 __ 3GO 360 

" "M" ' = ~s~ ~~ T" 
!__ L I 

"' M~"S~~Y 

IfSis27JandYis365i: 

1 3 4 



M 82 1461 
M = 29J (approx.) 

The same figure also shows us the nodes where sun and moon have the 
same declination corresponding to the same R.A. Since eclipses can only 
occur if the moon is near a node, and when the sun's R.A. is the same as that 
of the moon (solar eclipse) or differs from it by 180 (lunar eclipse), we can 



104 Science for the Citizen 

calculate the time of eclipses by observations of this kind. Repeated records 
show that the position of the moon's node rotates along the ecliptic in accord- 
ance with the rule empirically discovered by the Sumerian priests (p. 46). 

By knowing the exact length of the sidereal month we can calculate the 
time of an occultation. A star or planet will hide behind the moon's disc at 
the day and hour when its R.A. agrees with that of the moon and its declina- 
tion does not differ from that of the moon by more than half the angular 
difference of the two edges of the latter. The angular diameter of the moon 
is almost exactly y (i.e. we have to turn a telescope through J to focus the 
two edges successively at the centre). To make such forecasts far ahead it 
is also necessary to take into account the rotation of the nodes. If we have 
tables of the moon's R.A. and declination on a given date at some stated 
longitude, we can compare the time of local noon with noon at the stated 
longitude by noting the interval which elapses between local noon and the 
time when the moon has the R.A. or declination tabulated for a particular 
hour. 

The principle which was refined for use in British navigation by Newton, 
underlies the so-called "method of lunar distance." It is referred to in a 
twelfth-century treatise by the monk Gerard of Cremona, who translated 
several Arabic versions of the Alexandrian astronomical works. Gerard (cited 
by Wright, his, vol. v, p. 83) states: 

When the moon is on the meridian, if you compare her position with that 
given in the lunar tables for some other locality, you may determine the differ- 
ence in longitude between the place where you are and that for which the lunar 
tables were constructed by noting the differences in the position of the moon 
as actually observed and recorded in the tables. It will not be necessary for you 
to wait for an eclipse. 

Portuguese ships, which carried Jewish astronomers schooled in the 
Arabic cartography, already practised this method in the fifteenth century. 
In an early sixteenth-century treatise on navigation elucidated by Professor 
E. R. G. Taylor, we find a reference to its use by the seamen of Dieppe. 
The passage is worth quoting, because it shows the eagerness with which 
mediaeval shipping everywhere made claim to astronomical science (cited 
from Geographical Journal, 1929): 

The cartographical work of Jean Rotz is typical of the French school of 
the fourth and fifth decades of the sixteenth century (e.g. Desceliers, Vallard); 
nor is his treatise on Nautical Science unique, save in its survival. . . . The 
title runs : "Treatise on the variation of the magnetic compass and of certain 
notable errors of navigation hitherto unknown, very useful and necessary to 
all pilots and mariners. Composed by Jan Rotz, native of Dieppe, in the year 
J542." ... In a long and flattering preface to the King, the writer says 
that he comes before him not empty-handed, but bearing a book and an instru- 
ment for his acceptance : the book composed for all those who wish to taste the 
pleasant fruits of Astrology and Marine Science. . . . The third part treats 
of the construction and use of the instrument, which the inventor calls a 
Differential Quadrant. Actually it is one of the precursors of the theodolite, 
and a very elaborate one. The large magnetic compass set in the horizontal 
circle suggests a marine origin, and it may be compared with the contemporary 



Pompey's Pillar 105 

instruments designed for taking horizontal bearings by such landsmen as the 
brothers Arsenius, which were orientated by tiny compasses inset on the 
margin. Like Waldseemuller's Polymetrum, Rotz' instrument could take a 
combined altitude and azimuth, but its prime purpose was for the accurate 
determination of the variation of the compass. . . . He gives two supposititious 
examples of longitude determination, as follows: "je dycts que le 15 e de 
Janvier 1529, moy estant dessus la mer voulus scavoir la distance de mon 
meridien au meridien dulme." At sunrise, then, which occurred at eight o'clock, 
he took his astronomer's staff "diet par les mariners esbalester" and measured 
the distance between the sun and moon, finding it 41. Two folios of compli- 
cated calculations follow, and finally, taking Ulm to be in 47 N., 30 20' E., 
the required longitude is found to be 180 E. of Ferro, i.e. somewhere east 
of the Moluccas. Using in the second example the moon and a fixed star, he 
says, "Moi estant sur la mer veulant scavoir la distance de mon meridien de 
Dieppe, premierement je rectifies mes ephemerides ou tables dalphonse au 
meridien de dieppe at puis je regard le vrai lieu de la lune pour ung certaine 
heure de la nuyt," namely ten o'clock, when he finds it 16 in Taurus, and 
determines its declination. He also finds the right ascension of the star Aldebaran 
"aprez toutes rectifications faictes destiez" 2 18' in Gemini, and its declina- 
tion 35 55' N. "Et notte ycy ung poinct cest quil est necessite que tu rectifies 
ton heure par en moyen des equations des heures mis aux tables dalphonse ou 
aux ephemerides." Again, a couple of folios are occupied by computations, 
and the longitude works out as 229 30' E., or somewhere in mid-Pacific. . . . 
This Dieppe seaman had no reason to complain of his personal reward at the 
King's hands. He was taken into the royal service, and described himself as 
"servant of the King" in the Boke of Ydrography (written in English), the 
preparation of which was his first official duty. At Michaelmas in 1542, he 
received a payment of 20, being one-half of an annuity of 40, then a very 
considerable sum, while on October 7th of the same year he was granted papers 
of denization for himself, his wife and children. 



CULTURAL FRUITS OF THE ALEXANDRIAN STAR MAPS 

The star maps of Hipparchus (c. 150 B.C.) differed in one particular 
from those which are usually used today. They showed the star's position 
(see Chapter IV, p. 220) in circles like circles of declination drawn parallel to 
the ecliptic instead of the equator, and meridians radiating from the pole 
of the ecliptic plane. When the plane of the ecliptic is used instead of the 
equator, the angles corresponding to right ascension and declination are 
called celestial longitude and celestial latitude. The reason for choosing the 
ecliptic was that the moon and planets all revolve very nearly in the same 
plane as the sun's apparent track. The reason for preferring the equator is 
first that right ascension and declination are related in a simple way to 
longitude and latitude on the earth itself, and second that the determination 
of right ascension and declination involves no elaborate calculation when 
the times and altitudes of meridian transits have been recorded. 

The first consideration was not obvious when star maps were introduced. 
Latitude and longitude were orginally devised to describe the relative 
positions of objects on the celestial sphere. It was a short step to the recog- 
nition that the earth itself can be divided into similar zones with simple 
relations to the fixed stars. Marinus of Tyre is credited with the preparation 



io6 Science for the Citizen 

of the first terrestrial map in which meridians of longitude and parallels 
of latitude were laid down. One immense cultural benefit of this was the 
extent to which it broadened man's knowledge of the habitable earth. You 
will appreciate this by comparing (Fig. 35) the world picture of the Greeks 
with that of Ptolemy (c. 200 A.D.), who garnered the fruits of Alexandrian 
astronomy in a work which usually bears the name of its Arabic translation. 
Jewish scholars, who sustained the traditions of the Moorish universities 
of Toledo, Cordova, and Seville, after the Christian conquerors had replaced 
the public baths by the odour of sanctity, handed on the Almagest to 
European navigators. 

At first sight it might seem strange that in exploration the achievements 
of the Alexandrians and the Arabs, who kept astronomy alive in the dark 
ages of Faith, were so small as compared with those of their pupils in the 
fifteenth and sixteenth centuries. For several reasons there was an inescapable 
lag between the theoretical equipment which Alexandrian science bequeathed 
and its full use in navigation. One is that the Alexandrians had no convenient 
portable instrument for recording time. On land, crude measurements of 
longitude could be made by lunar methods with the help of hour-glasses or 
water-clocks. At sea, methods of determining longitude known in antiquity 
could not be used. The need for seeking out new methods did not become an 
acute technological problem so long as a large part of the globe could still 
be explored by sailing close to the coast. 

In the great voyages associated with the names of the pharaoh Necho, 
and with the Carthaginian Hanno in antiquity, knowledge of latitude was 
adequate for the purpose of locating a place, because all the long distance 
expeditions of antiquity steered a northerly-southerly course along a coast- 
line. The same remark applies to the early expeditions of the Portuguese 
and Dieppe shipping guilds. The need to determine longitude became a 
real and acute one when the international exchange economy entered on 
its last phase at the end of the sixteenth century. Oarless ships then ventured 
out into the uncharted west far beyond the sight of land, equipped with 
wheel-driven clocks. To be sure, they were clumsy and inaccurate instruments 
according to our standards, yet an immense convenience when compared 
with hour-glasses. The ships of Columbus, Vespucci, and Magellan had to 
put into port to take a bearing in longitude. None the less, what observations 
they succeeded in recording introduced a new assurance into navigation 
and created the technological problem which in turn stimulated the develop- 
ment of optics, dynamics, and terrestrial magnetism. 

A passage in Hakluyt's voyages is worth quoting to emphasize the import- 
ance of the new orientation which transatlantic navigation involved. It 
occurs in a letter of Thomas Stevens from Goa in 1579 : 

Being passed the line, they cannot straightway go the next way to the 
promontory; but according to the winde, they draw alwayes as neere South 
as they can to put themselves in the latitude of the point, which is 35 degrees 
and a halfe, and then they take their course towards the East, and so compasse 
the point. But the Winde served us so that at 33 degrees we did direct our 
course towards the point or promontory of Good Hope. You know that it is 
hard to saile from East to West or contrary, because there is no fixed point in 



Pompey's Pillar 



107 



all the skie, whereby they may direct their course, wherefore I shall tell you 
what helps God provided for these men. There is not fowle that appereth, 
or signe in the aire, or in the sea, which they have not written, which have made 
the voyages heretofore. Wherefore partly by their owne experience . . . arid 
partly by the experience of others, whose books and navigations they have, 
they gesse whereabouts they be. 

By this date longitudes are mentioned frequently in the records of English 
expeditions contained in Hakluyt's pages. Fifty years earlier it is clear that 
English seamen were only familiar with the determination of latitudes. The 
ensuing passage from a letter written in 1527 by Robert Thorne to Ley, 




FIG. 54. THE ASTRONOMER MAPPING THE HEAVENS 

A woodcut from the title page of Messahalah, De Scientia motus orbis> printed by 

Weissenburger in 1504. 

ambassador of Henry VIII to the Emperor Charles, points to considerations 
of high politics conspiring with mere convenience to promote the study of 
longitude : 

Now for that these Islands of Spicery fall neere the terme and limites 
betweene these princes (for as by the sayd Card you may see they begin from 
one hundred and sixtie degrees of longitude, and ende in 215) it seemeth all 
that falleth from 160 to 180 degrees should be of Portingal: and all the rest 
of Spaine. And for that their Cosmographers and Pilots coulde not agree in 
the situation of the sayde Islandes (for the Portingals set them all within 
their 180 degrees and the Spaniards set them all without) and for that in 
measuring, all the Cosmographers of both partes, or what other that ever 
have bene cannot give certaine order to measure the longitude of the worlde, 



io8 Science for the Citizen 

as they doe of the latitude; for that there is no starre fixed from East to West, 
as are the starres of the Poles from North to South, but all moveth with the 
mooving divine; no maner can bee founde howe certainely it may bee measured, 
but by conjectures, as the Navigants have esteemed the way they have gone. 

The influence which Ptolemy's work exercised in the period which immedi- 
ately preceded the Great Navigations may be illustrated by a passage from 
Roger Bacon's Opus Majus^ cited by Professor E. R. G. Taylor*: 

Again in the second book of the Almages he wrote that habitability is only 
known in respect of a quarter of the earth, namely that which we inhabit. . . . 
Whence it follows that the eastern extremity of India is a great distance from 
us and from Spain, since it is so far from the Arabian Gulf to India. . . . And 
so in the two quarters beyond the equinoctial (i.e. the equator) there will also 
be much habitable land. 

In Pierre d'Ailly's Imago Mundi the passages which greatly influenced 
Columbus are mainly taken from Roger Bacon. In Roman Alexandria such 
possibilities could be entertained with a light heart. The temper of Bacon's 
times was different. To quote from Professor Taylorf once more, 

many thinkers accepted the view found in the DC Situ Orbis of Pomponius 
Mela, namely, that a habitable region, girt about by ocean, was to be found in 
each of the temperate zones, separated by a torrid zone impassable because of 
the heat. Supposing this to be the case, and that the Antiothones actually did 
exist, then these people were for ever cut off from the means of Salvation: it 
was for this reason that St. Augustine, in an oft-cited passage, condemned such 
a view as heretical. 

Another fact in the everyday life of ancient times made it difficult to 
exploit the discoveries of Alexandrian astronomers to the fullest extent. 
Some celestial phenomena, e.g. the precession of the equinoxes or the 
motion of the moon's nodes, cannot be detected during short periods of 
observation without very accurate instruments such as we possess today. 
Although the first of these was discovered by Hipparchus, there were many 
others which escaped attention till modern times. Hence, astronomical tables, 
based on the principles we have dealt with so far, were liable to become out 
of date, i.e. inaccurate, in a comparatively short time. The older almanacs 
stood in need of constant revision. The introduction of printing into Europe 
on the threshold of the Great Navigations made it possible to maintain a 
fresh supply of almanacs for nautical use,^: besides increasing the proportion 
of people who were able to use them and distributing the results of the latest 
calculations far and wide. Meanwhile the first printing presses were also 
busy with the distribution of commercial arithmetics which were replacing 
the laborious and devious methods of Greek arithmetic by the simpler 
modern system derived from the Arabs. 

Two other cultural by-products of the Alexandrian star maps are of very 

* Scottish Geographical Magazine, vol. xlvii, 1931, pp. 215-17. 

t Ibid., vol. xlvii, 1931, p. 79. 

f The following figures give the publication of almanacs for the years stated : 

1448-58 2 1468-78 .. ..44 

1458-68 1 1479-88 . . . . 96 

14S9-98 110 



Pompey's Pillar 



109 



great significance for the subsequent advance of human knowledge. The 
method of Hipparchus essentially corresponds to the kind of graphical repre- 
sentation which mathematicians call polar coordinates, and flat maps of 
longitude and latitude essentially represent the more familiar way of plotting 
a graph in "Cartesian coordinates." From the Almagest mediaeval Europe 
learned graphical methods. In the fourteenth century Oresmus was giving 
lectures at Prague on the use of "latitudines and longitudines" to show the 
track of a moving point. Cartesian geometry emerged from the same social 
context as modern cartography. Alexandrian astronomy also precipitated a 
more decisive departure from the sterile tradition of Platonic geometry. From 
Euclid, the first teacher of Alexandrian mathematics, the Alexandrians had 




FIG. 55. ALBRECHT DURER'S WOODCUT OF THE SOUTHERN CELESTIAL HEMISPHERE 
FROM THE GEOGRAPHICAL WORKS OF JOHANN STABIUS, 1515, PROFESSOR OF MATHE- 
MATICS AT VIENNA AND ASTRONOMER TO THE EMPEROR 

Note Orion containing the bright stars Regel and Betelgeuse, Canis minor with the 
bright star Procyon, Canis major with the brightest star Sirius, are visible in the 
latitude of London. Argo with Canopus, the second brightest star in the heavens, is not. 

received Plato's idealistic doctrine that mathematics should be cultivated as 
an aid to spiritual refinement. The first measurements of celestial distances 
sufficed to show the clumsiness of the Platonic methods. The stubborn realities 
of the material world forced them to refinements which resulted in cultivating 
new methods. Hipparchus compiled the first trigonometrical table a table of 
sines. Thenceforth Plato's geometry was an anachronism. With the aid of 
six or, at most twelve, of its propositions, we can build up the whole of trigo- 
nometry and the Cartesian methods. Unfortunately the curriculum of our 
grammar schools was designed by theologians and politicians who believed in 
Plato. So we continue to teach Euclidean geometry for the good of the soul. 
Since few normal people like what is good for them, this makes mathematics 



no 



Science for the Citizen 



'A 




lyp 



^ 
^3 

4! 



otsmut 



90' 



.if A 



(i) sin A =---- cos (90 - A) 
cos A sin (90 A) 




P- 



Base of .90 -A 



(in tan A - 



sin A 

- r- 
cos A 



' 



- ~ - = ' x T 
// h // 



h \ 
b) 




sin A = .". p == c sin A 
sin C = :. p ~ a sin C 

<2 

sin A = a sin C 
sin A sin C 



sin A = 



asinC 



^ a 2 -(b ~ d) 2 

.'. c- - d z ^ a* - W 4- 2bd - d 2 
.'. c* - a z - b 2 + 2bd 

Since cos A = d ~ c 3 i.e. J = c cos A 
^> 2 -f 2&c cos A 
a 2 = 2bc cos A 



c z = a 



c- + 6 2 - a 2 
COSA ^ 2^ 



FIG. 56. THREE FUNDAMENTALS OF TRIGONOMETRY 

If a right-angled triangle is completed about an angle A, the ratios of its sides severally 
(called sine A, cos A> and tan A) are given in tables. With the aid of these the remaining 
dimensions of any triangle can be found by applying the four rules of the figure if 
we know the length of one side and the size of two of its angles, or the length of two 
sides and the angle included, or the lengths of all three sides. 



Pompey's Pillar 



in 



a most unpopular subject and effectually prevents the majority from ever 
finding out the immense usefulness of mathematics in man's social life. One 
result of not understanding how mathematics can be usefully applied is that 
many people do not realize when it cannot be usefully applied. Hence the 
delusion that a subject is entitled to rank as a science when it contains formulae. 

THE SIZE OF THE MOON AND ITS DISTANCE FROM US 

Trigonometry developed in connexion with the attempt to solve problems 
which had a very far-reaching influence upon human belief. The Alexandrians 
made the first estimates of the distance of the moon and the sun from the 
earth. Their measurement of the sun's distance involved that of the moon. 

The principle involved in finding the distance of the moon is essentially 




FIG. 57. FINDING THE HEIGHT OF A CLIFF BY THE METHOD OF PARALLAX 
h ^ h 



tanP = 



+ d 

* 
tanP 



tan Q 



\tan P tan 



:.h 



tanQ 
tan P tan Q 



tan Q tan P 

N.B. The angle C is the parallax of the top of the cliff with reference to observers 
at A and B. 

the same as that used for finding the distance of a mountain peak. The 
method used for terrestrial objects, when we have tables of sines or tangents 
at our disposal, is shown in Fig. 57. The observer records the difference in 
direction of the object as seen from two situations at a measured distance 
apart. This difference of direction measured by the angle C is called the 
parallax of the object. So far we have assumed that the different angular 
positions of the celestial bodies, when observed at the same time in different 
places, are entirely due to the curvature of the earth, the direction of a star 
being fixed. In other words, it does not matter where we determine the R.A. 
and declination of a visible star. This is what we should expect if the stars 
are very remote. The nearest stars are too far away to make a difference of 
a thousandth of a degree. This is not true of the moon. Two simultaneous 



112 Science for the Citizen 

determinations of the moon's R.A. by observers situated at longitude 90 
apart on the equator, or simultaneous determinations of declination by one 
observer at the equator and another observer at either of the poles, can differ 
by almost exactly one degree; and this difference is detectable with a com- 
paratively crude instrument. In an age when the study of the moon's be- 
haviour was important for regulating social festivals or night travel, and the 
study of solar eclipses and occultations had begun to assume a practical as 
well as a magical significance, differences of the moon's direction at different 
stations did not escape detection. They involve an appreciable correction in 
all calculations of the moon's position. The moon's position relative to a more 
distant celestial object is not quite the same as seen from two widely separated 
stations simultaneously. Consequently crude methods of determining longi- 
tude based on solar eclipses, occultations or lunar distances, can be somewhat 
inaccurate, unless a correction is made for the moon's parallax. 

If we know the earth's radius we can calculate the distance between two 
observers at known latitudes and longitudes, and the parallax of the moon 
is all that is required to deduce its distance from the earth. The best method 
is to make simultaneous observations of the moon's zenith distance, when 
it crosses the meridian, at two stations on the same meridian of longitude 
at latitudes as far apart as possible. If the moon's rays were absolutely 
parallel, the south zenith distance at a given north latitude L when the 
moon's true declination (i.e. elevation from the plane of the equator) is D, 
would be (Fig. 58): 

*-= L-D 

Actually the observed zenith distance Z is a little in excess of this by an 
angle p (called the geocentric parallax at L) representing the difference of 
direction of the moon as it would be seen simultaneously at L and at the 
earth's centre if we could get there. Thus, 



We have already seen (Fig. 28) that if the rays from a celestial body are 
perfectly parallel at two different latitudes, the difference of its zenith 
distances at two places is the difference of their latitudes, i.e. 

JL*-i % ~~~~ ** 2 1 

If one latitude is south of the equator, its sign is negative ( L 2 ) and if 
the moon is seen south of the zenith at the northern latitude and north of 
the zenith at the southern, its zenith distance at the former is negative 
( - ZJ, and at the latter, positive (+ 2 ), so that 



The quantity (p v + p 2 ) is the moon's parallax P with reference to the two 
stations, so : 

P = (Z l + Zg) - (L, + L,) 



Pompey's Pillar 

Turning to Fig. 59, you will see that if both stations are on the same longitude 
and both observations are made at the same time, the moon's distance (d) 
from the earth centre is the diagonal of a four sided figure of which all four 
angles and two sides (equivalent to the earth's radius or approximately 
4,000 miles) are known. Hence the quadrilateral can be constructed. Its 
remaining sides and the diagonal can be calculated by the fundamental 
relation (Fig. 56) between the sides and angles of any triangle, i.e. 

sin A sin B sin C 




FIG. 58 

The moon's geocentric parallax. If the moon's rays were truly parallel its direction at 
latitude -f L would be the same as at the earth's centre, i.e. its zenith distance would 
be z = L D. Actually (upper figure) the observed z.d. at transit for an observer 
at latitude L is greater by the angle p (geocentric parallax at L). This cannot be directly 
measured, but the sum p t 4- p z of the geocentric parallaxes of the moon for observers 
at two different latitudes (+ LI and L 2 ) on the same longitude is 

(Z, + Z 2 ) - (L, + L a ) 

In this way the mean distance of the earth's centre from the moon is found 
to be approximately 240,000 miles, as shown in the legend attached to 
Fig. 59. Hipparchus gave the distance of the moon as between 67 and 78 
radii (i.e. mean distance 72 radii or 280,000 miles). This is a tolerable 
approximation. His estimate of the sun's distance based on the study of 
eclipses was 13,000 earth radii or 51,000,000 miles, which is much too 
smallj the correct distance being 92,000,000 miles. 

The calculation of the moon's radius is easily deduced from this. The only 
new information required is the moon's angular diameter, i.e. the angular 



Science for the Citizen 

difference between the two edges of the moon. This is almost exactly half a 
degree. As shown in Fig. 60, the moon's diameter is therefore a little over 
2,000 miles. It is almost exactly 2,160 miles, or three-elevenths that of the 
earth. 




Since BC = r = DC 

ZlCBD - x - Z.CDB 

y = 180 - Zj - X 



ZBAD - P 

ZlBCD - L, -h L 2 

ZCBA - i8o - Zi 

/.CD A = 180 - Z 2 
BC - r - DC 




(i) In the A BCD, /_x, zlBCD and CB are known, and 
sin x sin BCD 



BD is known 



BD 



(ii) In the A ABD, /_y, /.BAD and BD are known, and 
sin BAD __ sin y 

BD "~ "AD" 
.*. AB is known 

(iii) In the A ABC, /.ABC, BC (= r) and AB are all known, 

.'. AC 2 - AB 2 + BC 2 - 2AB . BC cos ABC (See Fig. 56) 
.'. AC ( d) is also known. 

FIG. 59 

The determination of the moon's distance, from the parallax of the moon with reference 
to two observers at different latitudes and the same longitude, as in Fig. 58, can be 
deduced from a simple geometrical construction. 

One of the earliest Alexandrian astronomers, Aristarchus, made a rough 
estimate of the ratio of the sun's distance to that of the moon (Fig. 61). 
The method he used depends on the fact that the half moon at first quarter 
occurs a little earlier than the time when the moon's direction differs from 
that of the sun by 90 .> and the half moon or third quarter occurs a little later 



Pompey's Pillar 

than the time when the moon's direction differs from that of the sun by 
270. Aristarchus calculated that the ratio of the distances of sun and moon 
from the earth is about 20 : 1. In reality it is nearly 400 : 1. The reason 
for the inaccuracy is the extreme difficulty of ascertaining when exactly 
half of the moon's face is visible without the aid of the telescopic observations 
on the moon's mountains. However, the estimate of Aristarchus was a vast 
step beyond the boldness of Anaxagoras, who startled the court of Pericles 
by the suggestion that the sun might be as large as the whole mainland of 
Greece. In using mathematics to measure the heavens, Alexandrian astronomy 




-^rriA 





77100715 

\ r&diiis 
v 



sin IA moon's radius moon's distance 
.*. "r =- sin | A \ 240,000 

By observation the moon's angular diameter, 

A - :n'r>" 

.". iA 
.". Moon's radius 



O n 16' approximately 
--= 240,000 x sinO 16' 
= 240,000 x 0-0047 
= 1,1 30 miles 
=- 2,260 miles. 



Diameter 

FIG. 60. THE MOON'S DIAMETER 

disclosed a vision of grandeur concealed from the world view of Plato's 
idealism. Alexandrian sky measurement was the death knell of Plato's 
cosmology and of the ancient star god religions. 

The sun's parallax is too small to be measured without the aid of good 
telescopes at observation stations separated by long distances. An alternative 
method which does not require the use of a telescope depends on observa- 
tions of eclipse phenomena. If the time of greatest duration of a total lunar 
eclipse is known from repeated observations, the distance, radius and synodic 
period of the moon, and the earth's radius, give us all (Fig. 18) the necessary 
data for measuring the breadth of the earth's shadow cone where it is inter- 
cepted by the moon's orbit. In a solar eclipse the moon's edge just coincides 
with that of the sun, which therefore has approximately the same angular 



Science for the Citizen 

diameter. So the sun's radius and the earth distance are in the same ratio as 
the moon's radius and earth distance. The ratio of the earth's radius to the 
length of its own shadow cone is the same as the ratio of the sun's radius to 
the combined length of the earth's shadow cone and the sun's earth distance. 
Since the length of the shadow cone is easily calculated by similar angles 
if its breadth at any known distance from the earth is given, all the requisite 
data are supplied by eclipse measurements and the moon's parallax. Actually 




Earth 



SUN 



(Second "Method.) 



FIG. 01. How ARISTARCHUS TRIED TO MEASURE THE SUN'S DISTANCE 

Exactly at the first quarter or third quarter, more than half of the face of the moon is 
seen. Half moon occurs when the moon has still to move through an angle B before 
it is exactly at the first quarter. Aristarchus found that the delay between half moon 

and first quarter was six hours, so, if a lunar month is taken as 28 days, B = ( ...... * --} 

\4 X 2o/ 

about 3. In the half-moon position, lines joining the sun and the earth to the moon 

meet in a right angle. So a = B = 3 and sin 3 = g^ ^anc from moon 

Earth s distance from sun 



this method, which is referred to in the ensuing passage from Snyder's 
book, is extremely inaccurate because small errors of observation lead to large 
differences in the ratios determined from them : 



The problem of the shadow cone, as is clear from the pages of Ptolemy, 
had been worked out by Hipparchus, apparently with great precision, but 
with the strange result of confirming the calculations of Aristarchus. He, 
too, found the distance of the sun about twenty times that of the moon, or 
from 1379 to 1472 half diameters of our globe. Ptolemy, a couple of centuries 
later, tries his hand at the matter, but with no better success; indeed, he reduces 
the distance to 1210 such half diameters. But with three distinct methods 
leading identically to the same result, there could now be little question of 



Pompey's Pillar 117 



their truth. There seems, indeed, to have been no question for another seven- 
teen centuries, and until Galileo and the telescope had come. Hipparchus' 
method it is generally so styled is reproduced with new proofs, but similar 
estimates, in the De Revolutionibus of Copernicus, A.D. 1543. The Theorem of 
Hipparchus gave not merely the relative but also the absolute measures of the 
solar and lunar distances, hence a direct measure of their size. Cleomedes 
tells us that Hipparchus computed the sun's bulk at 150 times that of the 
earth; Ptolemy made it 170 times. But Aristarchus, by what method he does 
not state, figured the diameter of the sun at between six and seven times that 
of the earth, hence about three hundred times its bulk. He sets the moon's 
diameter at one-third that of the earth an error of but one-twelfth ; admirable, 
if yet imperfect approximations. The march of the mind had begun! . . . 
There is, in an oddly-jumbled work, Opinions of Philosophers, attributed, with 
slight probability, to familiar old Plutarch, a paragraph which says that 
Eratosthenes had engaged the same problem. True to his love of concrete 
measures, he gives the distance of the moon at 780,000 stadia, of the sun at 
804,000,,000 stadia. Marvellous prevision of the truth! For though he makes 
the distance of the moon only about twenty earth radii too small by two- 
thirds of the reality his figure makes the sun's distance 20,000 radii, which, as 
nearly as we may estimate the stadium, was practically the distance that, after 
three centuries of patient investigation with micrometers and heliometers, is 
set down as the reality. ... It is with a deepening interest, bordering even 
upon amazement, that we find yet another great investigator of antiquity 
announcing similar but quite distinct estimates. This was Poseidonius, the 
teacher of Cicero and of Pompey, one of the most contradictory of characters, 
now seeming but a merest polymath, now one of the most acute and original 
thinkers of that ancient day. We have already noted that his measure of the 
earth, adopted by Ptolemy, was the sustenance of Columbus. He had closely 
studied the refraction of light, and gives us a really wonderful calculation as 
to the height of the earth's atmosphere. In the pages of Cleomedes we learn 
that he equally attempted to establish the distance of the stars. He puts the 
moon at two million stadia away, the sun at five hundred million! This, on his 
earlier estimate of the earth's diameter, would place the moon at 52 radii 
of the earth, which would be nearer than the computations of Hipparchus. 
It would make the sun's distance 1.3,000 radii. If we take his later figure 
(180,000 stadia), the distance would become 17,400 radii, an estimate which, 
considering the necessarily wide limits of error, does not differ greatly from 
that of Eratosthenes, and equally little from the truth. Compare it with the 
thirteen hundred radii of his forerunners! Compare it with the notions of 
Epicurus, almost his contemporary, a very wise and large-minded man in his 
way, who yet believed that the sun might be a body two feet across ! 

The importance of science in the everyday life of mankind means more 
than making it possible for us to organize material prosperity. The advance of 
science liberates mankind from beliefs which sidetrack intelligently directed 
social effort. In the phraseology of the materialist poet Lucretius, science 
liberates us from the terror of the Gods. At a later date the scientific study 
of the heavens was destined to challenge the authority of custom-thought 
in a more spectacular context, by discrediting the biblical doctrine that the 
sun revolves round the earth. The view which is now held was recognized 
as a possible alternative by the Pythagorean brotherhoods and advocated 
by Aristarchus. For a very good reason it was rejected by Hipparchus, on 



Science for the Citizen 

whose teaching the Almagest is largely based. He did so, because no par allay, 
of the sun could be detected. 

Observations on the fixed stars made at different localities at the same 
time yield no appreciable difference in R.A. or declination. In other words, 
the fixed stars have no measurable geocentric parallax. Since even the moon's 
geocentric parallax is never a very big quantity, this would be quite in- 
telligible on the assumption that they are much farther away than the moon. 
Aristarchus estimated that the sun was about 20 times as distant as the 
moon from the earth. When the moon's distance was found to be about 60 
times the earth's radius, it appeared that the diameter of the earth's orbit 
must be more than a thousand times the diameter of the earth. The 
difficulty did not end here. If the earth moves round the sun, it must be 
nearer to any particular star at some seasons than at others. It still seemed 



annual pat+allax oi 

a- fixd star 




FIG. 62 

hard to believe that the stars are too far away to show any change of 
apparent direction when the earth is at opposite ends of its orbit (as in 
Fig. 62). With any instruments available before the nineteenth century, no 
annual parallax of the fixed stars could be detected. The refined methods 
we now have show that the heliocentric or annual parallax (P in Fig. 62) 
is never as much as one four-thousandth of a degree (0*18" for the nearest 
star, Proxima Centauri). According to the only test which was known to the 
ancients, the doctrine of Aristarchus failed. On the evidence which their 
instruments yielded, the founders of the Ptolemaic system were right in 
putting the earth at the centre of the star map, where it is still convenient to 
leave it for the purposes of navigation and scientific geography. 

Nowadays we often read dogmatic statements about the discredit into 
which Newton's system has fallen, and there is danger of losing sight of 
what is permanent in any useful hypotheses. Scientific hypotheses that yield 
a useful explanation of some facts of experience, are not relegated to the limbo 
of superstition whenever we discover as we constantly do discover 



Pompey's Pillar 



119 



new facts which they do not explain. The Ptolemaic astronomy which pro- 
vided the technological basis of the Great Navigations is still the most 
convenient representation of the apparent movements of the sun, moon, and 
fixed stars. It is the world view of the earth-observer, and as such remains the 
basis of elementary expositions on nautical astronomy after nearly two 
thousand years. Its greatest disadvantage which eventually brought it into 
disrepute is that it can give no reliable means of ascertaining the motions of 
another class of heavenly objects the planets, which are not fixed in the 
heavens like the stars. If we use the methods of Chapter IV to plot the move- 




* - 

Scorpio 



Virgo 

Libra, 
FIG. 63. THE TRACK OF VENUS IN 1894 



ments of a planet like Venus on the star map shown in Fig. 51, they do not 
conform to any simple rule. The nearer planets wander about the heavens, 
each in a seemingly capricious track, with no evident geometrical pattern. 
Their motions are easier to calculate if we put ourselves in the position of 
a sun observer. 

We shall see later how Copernicus furnished the germs of a comparatively 
simple account of the way in which the planets move by discarding the 
belief that the earth is the centre of the universe. In the absence of satisfactory 
evidence for the annual parallax of the fixed stars (before about A.D. 1830), 
it is highly doubtful whether the usefulness of the Copernican doctrine 
for calculating the positions of the planets would have sufficed to overcome 



120 



Science for the Citizen 



the combined authority of the Book of Joshua and the Almagest, had there 
been no new sources of information to discredit belief in the fixed position of 
the earth at the hub of the heavenly wheel. How the ancient astronomers 
attempted to explain the vagaries of the planets or "wanderers" of the heavens 
may best be dealt with when we come to the Copernican hypothesis. We shall 
first see how two devices prepared the way for its universal acceptance. The 
introduction of the telescope and the pendulum clock in the beginning 
of the seventeenth century registers an eventful phase in the conquest of 



Pole Star 




FIG. 04. THE MOORISH SUNDIAL 

distance and time. It is hardly too much to say that, with one exception, no 
other technical advances contributed so much to change the world picture 
of mankind between 4241 B.C. and A.D. 1800. 

In the period which intervened, Arab astronomers and mathematicians, of 
whom Omar Khayyam was one of the most illustrious, blazed the trail for 
the new world outlook by more accurate measurements of the positions of 
the planets. A far more important Arabic contribution lay in devising simple 
rules of calculation. The use of mathematics in Alexandrian science was 
essentially pictorial. Mathematics was applied almost exclusively to astro- 
nomical data. The trigonometry of Hipparchus and the algebra of Diophantus, 
the last Alexandrian mathematician of importance, were imprisoned within 
the obscure number symbols of the Greek alphabet. By discarding the 



Pompey's Pillar 



121 



alphabet and adopting the Hindu sunya or zero, as we call it, the Arab 
mathematicians equipped us with an arithmetic which could accommodate 
a growing body of scientific measurements, and with an algebra which has 
extended the application of mathematics far beyond the confines of geometry. 
Located in sunny regions, the ingenuity of the Islamic culture was not 
compelled to perfect mechanical devices for recording the time of day. The 
contribution of the Arab astronomers to the technique of recording time is 
now relegated to the status of a garden ornament. The ancient shadow clocks 




= sin Z, tank 



FIG. 65. How TO CALIBRATE THE SUNDIAL 

A solid model of the sunbeam, style, and shadow, can be made by folding the upper 
figure SPQSR. The angle H is the shadow angle with the meridian; L the latitude 
of the place and the inclination of the style to the horizon plane ; and h is the sun's 
hour angle. The figure shows that 

tan H = sin L tan h. 

Thus to mark off the angle H corresponding to two o'clock (p.m.) or ten o'clock (a.m.), 
i.e. when h = (15 x 2) = 30, at latitude 51 we have 

tan H = sin 51 tan 30 

= 0-7771 X 0-5774 = 0-4487 

/. from tables of tangents H = 24-15. 

did not record a constant hour at different seasons. The Arab sundial (Fig. 
64) with its style set at an angle equivalent to the latitude and pointing due 
north marked out hours which were mechanically equivalent throughout 
the year. The only lasting importance of this invention, the theory of which 
is explained in Fig. 65, lies in the fact that the first mechanical clocks were 
checked by comparison with the readings of the dial. 

The state of geographical knowledge available for navigational purposes 
in the time of the Crusades is indicated in the following passage from Wright's 
book already cited : 



122 Science for the Citizen 

That the Saracens also were interested in the more strictly mathematical 
aspects of astronomical geography is emphatically proved by the fact that they 
undertook actually to measure the length of a terrestrial degree and thereby 
to determine the circumference of the earth. Some knowledge of this great 
work came to the Western world in our period through translations of the 
Astronomy of Al-Farghani. . . . Astrology also necessitated this type of 
investigation. In order to cast a horoscope one must know what stars are over- 
head at a particular moment; and, to ascertain this, one must know latitude and 
longitude. In the Arabic astronomic works there occur rules for determining 
positions and tables of the latitudes and longitudes of places throughout the 
world. One of the most practical results of Arabic investigations in this field 
was a reduction of Ptolemy's exaggerated estimate of the length of the Medi- 
terranean Sea. The Greek geographer gave the length as 62, or about half 
again too long. Al-Khwarizmi cut this figure down to about 52 , and, if we 
are right in our interpretation of the available data, Az-Zarqall still further 
reduced it to approximately the correct figure, 42. . . . The results of these 
corrections became known in the medieval West. The Moslems, as a general 
rule, measured longitudes from the prime meridian which Ptolemy had used, 
that of the Fortunate Islands (now the Canaries), situated in the Western 
Ocean at the westernmost limit of the habitable earth; but individual writers 
came to make use of another meridian farther west, a meridian destined 
to become known to the Christian World as that of the True West, as distin- 
guished from the supposed border of the habitable West. Abu Ma'shar, on 
the other hand, referred his prime meridian to a fabulous castle of Kang- 
Diz a far to the east in the China Sea . . . there is absolutely no doubt that 
methods of finding latitudes and longitudes were well understood in theory 
and were sometimes put to practical use. Rules are given for finding latitude 
in Az-ZarqalPs Canons, in Plato of Tivoli's translation of the Astronomy 
of Al-Batt i ni, and in many other astronomical and astrological treatises. 
Two principal methods were recommended. You may either measure with the 
astrolabe the altitude of the sun above the horizon at noon at the spring or 
autumn equinox and find the latitude by subtracting this angle from 90 or 
you may measure the altitude of the celestial pole above the horizon, which is 
the same as the latitude. As to longitude, the fact that there are differences 
in local time between points east and west of each other was recognized and 
clearly explained by several writers of our age. The Marseilles Tables give a 
rule for finding longitude by the observation of eclipses. Roger of Hereford 
indicates that he himself, by observing an eclipse in 1178, ascertained the 
positions of Hereford, Marseilles, and Toledo in relation to Arin, the world 
centre of the Moslems. 

The Marseilles tables mentioned in the foregoing passage seem to be the 
earliest astronomical tables in north-western Europe. Referring to their 
origin, Wright says : 

Preserved in a twelfth-century manuscript of the Bibliotheque Nationale 
is a set of astronomical tables for Marseilles dating from 1140, the work of a 
certain Raymond of Marseilles. The Canons, or introductory explanation, of 
these tables are drawn largely from the astronomical Canons of Az-Zarqall; 
the tables are an adaptation for the meridian of Marseilles of the Toledo 
Tables. Both Az-Zarqalfs Canons and the Toledo Tables, with their modifi- 
cations like the Marseilles set, contained not a little incidental material of 
importance from the point of view of astronomical geography, including a 



Pompey's Pillar 



123 




FIG. 66. WALDSEEMULLER'S POLYMETRUM, FROM REISCH'S "MARGARITA PHILO- 

SOPHICS," 1612 

(From "A Regional Map of the Early Sixteenth Century" by Professor E. G. R. Taylor, 
Geographical Journal, Vol. LXXI, No. 5, May 1928.) 

list of cities with their latitudes and longitudes derived ultimately from Al- 
Khwarizmi. That this material enjoyed wide popularity during our period 
and later is proved by the existence of a large number of manuscripts. One 
of the translations of Az-Zarqalfs Canons was done by the hand of the famous 
Gerard of Cremona. 

In the fifteenth century Portuguese and Spanish shipping enjoyed the 
benefit of expert pilots schooled in the teaching which had been salvaged 
from the wreckage of the Moorish universities in the Peninsula. The growth 



124 Science for the Citizen 

of the printing industry placed the fruits of this knowledge at the disposal 
of sea captains with less erudition. Simple instructions involving no elaborate 
calculations like the "Regiment of the Pole Star" (Fig. 67) could now be 
put into the hands of all who could read. An account of these early pilot books 
is given by Professor Taylor in an article {Geographical Journal^ 1931) from 
which the following passages are cited : 

The earliest type of Seaman's Manual was the Rutter or Pilot Book, con- 
taining details of landmarks, anchorages, shoals, bottoms, watering-places, and 
so forth, which goes back to the Ancient Greeks, and probably to their Phoeni- 
cian predecessors. Such books belong to the period of coastwise sailing, when the 
lode-line was the sole navigating instrument other than the shipmaster's eyes. 
These Pilot Books have never been superseded, but from time to time as the 
art of Navigation developed, they have been supplemented. The earliest supple- 
ment was the Pilot Chart, and the date of its introduction is quite unknown: 
the oldest extant example, drawn about A.D. 1300, is so perfect and so stylized 
that it must have had a very long past history. . . . The same period brought 
a further innovation: the astronomers of the Western Mediterranean devised 
simple instruments and simple methods whereby the Stella Maris or Tramon- 
tane could be employed to fix the ship's position in latitude in addition to 
indicating the North. The development of this aspect of navigation, and the 
progress of exploration across the equator, led eventually to the introduction 
of the determination of position by the noon-tide altitude of the sun, which 
demanded a calendar and a set of tables, and thus an elementary forerunner 
of the Nautical Almanac became part of the Seaman's Manual. Even the 
simplest instrumental fixing of the ship's position, however, required some 
knowledge of astronomy, and consequently a fourth section was added to the 
Manual, which took shape as a preface dealing with the Earth as a Sphere 
and the Heavenly Bodies. By the early sixteenth century it had become usual 
for the Theory of the Sphere and the Tables and Rules for fixing position 
to be bound up together as the Navigating Manual proper, while the ship- 
master or pilot provided himself with the particular Pilot Book or Rutter, and 
the particular Chart which he needed. There were, however, interesting 
exceptions to this rule. . . . and a remarkable work on Navigation of the 
Fourteenth Century is extant which contains the rudiments of all four part 
of the complete Manual in a single volume. This is the rhyming Florentine 
example, known by the title of its first section on the globe as "La Sfera," 
which is ascribed to Gregorio Dati. The second part, dealing with the rules 
for navigation and finding position, is naturally very crude at this date, 
while the chart (the third part) is drawn in sections, which are placed in 
the margins of the appropriate portions of the Rutter (the fourth part). . . . 
The Italian manuscript entitled "Brieve conpendio de larte del navegar," 
in the Society's Library, contains three out of the four parts of the complete 
Manual, namely, the section on the Sphere, the Rules and Tables, and the 
Chart; only the Rutter is absent, and certain supplementary notes and tables 
are inserted instead. ... It was written by an Italian pilot, Battista Testa 
Rossa, whose patron was El Magnifico Marco Boldu, a member of one of 
the great ruling families of the Republic of Venice. It is dated 1557, when 
Battista had settled in London. . . . The section on the Sphere is of the 
simplest possible type, consisting of necessary definitions only, namely, of 
altitude, degree, horizon, zodiac, equinoctial line, declination, Circles, Poles, 
Tropics, seasons, longitude, latitude, parallels, meridians, hemisphere, zenith 



Pompey's Pillar 



125 




FIG. 07. THE REGIMENT OF THE NORTH POLE 

The star Polaris is only 1 off the true pole today. In mediaeval times Polaris revolved 
in a tiny circle 3 from the celestial pole as did a Draconis, when it was the guiding 
star of the Pyramid builders. Hence latitude by the altitude of the Pole Star might 
exceed or fall short of the true value by 3. The pilot books gave instructions for 
making the necessary correction by noting the hour angle of the star f$ in the Little 
Bear. The positions of Polaris relative to the true pole were charted for eight positions 
of the "guards" 0? and y) 3 so that naked eye observation sufficed to make a correction 
with an error less than half a degree. 

(Reproduced with Professor E. G. R. Taylor's permission, from her Tudor ^Geography, 
1 485-1583 (Methueri).) 



126 Science for the Citizen 

and centre. A brief exposition of the changing altitude of the Pole Star as the 
traveller passes from Alexandria to Iceland, and its explanation in terms of the 
rotundity of the Earth, concludes the section. The second section, on Rules and 
Tables, opens with the Regiment of the North Pole, i.e. the number of degrees 
(from 4 to 3 ) which must be added or subtracted from the observed altitude of 
the Pole Star, according to the position of the "guards," in order to obtain the 
latitude. "The guards" was the seaman's name for part of the constellation 
Ursa Minor, the two significant stars used as indicators being j8 and 7 . . . 
when the guards are in the east and the forward star (of the guards) is east of 
the North Star, then the Pole Star, as it is called, will be 1J below the Pole, 
and this must be added to the altitude, when the sum will be the height of the 
Pole above your horizon: when the guards are in the north-east, and the two 
stars of the guard, one with another by east, then the Pole Star will be 3 
below the Pole. . . . The "forward" or "fore" guard was the brighter of the 
two (i.e. )3 Ursae Minoris), and was sometimes called the "clock star," since it 
could be used to tell the time at night. Directions for making and using the 
balestilha or Jacob's Staff for these stellar observations are given, with a 
diagram to show how to add the scale to the Staff, while the quadrant and 
astrolabe are also mentioned. This Regiment of the North Star is one of the 
stereotyped Rules found in every Manual. Here it is followed, as in many other 
examples, by a rule to tell the time at night by the guards during each month 
of the year. . . . Now this makes it clear that the degree was taken by seamen 
as 70 Roman miles, equivalent to 64-4 English statute miles, a much less 
faulty figure than the 62 J miles (of 5,000 feet), or alternatively 60 miles, 
adopted by cosmographers. The method of derivation, too, is made plain. 
Battista gives 25,200 miles as the circumference of the Earth, which is clearly 
derived from Eratosthenes's measure of 252,000 stadia (made widely known 
to navigators through Sacrobosco's Sphera which was prefixed to the early 
Portuguese Manuals), by taking the equivalent of 10 sea stadia to a Roman 
mile. The Portuguese league of 4 miles was adopted by all seamen, as Columbus 
mentions, and as Jean Rotz, too, makes clear; but in Spain the degree of 11 \ 
leagues was presently superseded by that of 1 6 leagues, reckoning according to 
Spanish use, only 3 miles to a league. . . . The next section of the Italian 
Manual consists of a couple of pages and a diagram on the Regiment of the 
Southern Cross, and then follows the most important part of all, which finds a 
place in every book of Rules and Tables. This is the Regiment of the Sun, 
that is to say, the rules for taking the noon-tide altitude, and for using the 
accompanying Table of Declinations, which are given for four years, the last 
being the Leap Year. A Calendar was combined with the tables for the First 
Year. ... A few examples of latitude determination are worked out, and it 
may be noted that for solar observations the use of the astrolabe and not the 
balestilha is proposed. ... It was probably in 1558 that Stephen Burrough 
visited the Casa de Contratacion in Seville, there to be greatly impressed by 
the meticulous care with which Spanish pilots were trained and examined. . . . 
So far as chart-making is concerned, Tudor England had nothing to set against 
the magnificent achievements of the Portuguese, Spanish, Italian, and French 
school of Cosmographer-Pilots. 

A simple recipe analogous to the Regiment of the North Pole for use in 
southern latitudes is given in Hakluyt's Voyages. It refers to the Southern 
Cross, and is written by the mariner Edward Cliffe, who accompanied the 
consort ship of Drake in the 1577-9 expedition to the Strait of Magellan: 



Pompey's Pillar 127 

And thence we came along the coast being low sandie land till wee arrived 
at Cape Blanco. This Cape sheweth it selfe like the corner of a wall upright 
from the water, to them which come from the Northwardes : where the North 
Pole is elevated 20 degrees 30 min. And the Crociers being the guards of the 
South Pole, be raised 9 degrees 30 min. The said Crociers be 4 starrs, repre- 
senting the forme of a crosse, and be 30 degrees in the latitude from the South 
Pole; and the lowest starre of the sayd Crociers is to be taken, when it is directly 
under the uppermost; and being so taken as many degrees as it wanteth of 30, 
so many you are to the Northwardes of Equinoctial : and as many degrees as 
be more than 30, so many degrees you are to the Southwards of the Equinoctial. 
And if you finde it to be just 30 then you be directly under the line. 

THINGS TO MEMORIZE 

1. North Polar Distance = Z.D. of Pole Z.D. of star. 
Declination = 90 N.P.D. 

Latitude = altitude of Pole = 90 Z.D. of Pole. 
Declination = Z.D. -f Latitude. 
Latitude = Declination Z.D. 

2. Zenith distances south of the zenith, and latitudes and declinations 
south of the equator are negative. 

3. Sundial time + Equation of Time = Greenwich Mean Time. 

4. Star's R.A. = sidereal time of transit of star = (approximately) solar 

time of star's transit time of transit of T . 
Star's R.A. Sun's R.A. = local (solar) time of transit (approximately). 

5. See also Figs. 24, 43, 45 and 50. 

EXAMPLES ON CHAPTER II 
Use the graph in Fig. 32 for the "equation of time" correction. 

1. What are the sun's declination and right ascension on March 21st, 
June 21st, September 23rd, and December 21st? 

2. What approximately is the sun's R.A. on July 4th, May 1st, January 1st, 
November 5th? (Work backwards or forwards from the four dates given. Check 
by Whitaker.) 

3. With a home-made astrolabe (Fig. 22) the following observations were 
made at a certain place on December 21st: 

Sun's Zenith Distance Greenwich Time 

(South) (p.m.) 

74J 12.18 

74 12.19 

74 12.20 

73J 12.21 

73J 12.22 

74 12.23 

74J 12.24 

What was the latitude of the place? (Look it up on the map.) 

4. Find the approximate R.A. of the sun on January 25th, and hence at what 
local time Aldebaran (RJV. 4 hours 32 minutes) will cross the meridian on that 
night. If the ship's chronometer then registers 11.15 p.m. at Greenwich, what 
is the longitude of the ship? 

Examples continued on page 130 



128 



Science for the Citizen 



(Ahwc tf,i'k bhck hnc - Pre-HellentttLe ) SCIENCE IN 

>nun<ait Metallur^r Tvfcciioiic Bi 



PRECESSION OF 
EQUINOXES 




TIMl- 



Pompey's Pillar 



129 



ANTIQUITY (Bdcw AJuzk bkck line, 



UAL Navigation, ^Militarism 



Choral 



Route 



Th& 
Cafapufc PuUzy 



Aeoliplle 
CHART J 



Iirigatunt 



Screw & 
Water 
Lewi 




130 Science for the Citizen 

5. If the declination of Aldebaran is 16 N. (to the nearest degree) and its 
altitude at meridian transit is 60 from the southern horizon, what is the ship's 
latitude? 

6. The R.A. of the star a in the Great Bear is nearly 1 1 hours. Its declination 
is 62 5'N. It crosses the meridian 4 41' north of the zenith on April 8th 
at 1.10 a.m. by the ship's chronometer. What is the position of the ship? 

7. Suppose that you were deported to an island, but had with you a wrist 
watch and Whitaker's Almanack. Having set your watch at noon by observing 
the sun's shadow, you noticed that the star a in the Great Bear was at its lower 
culmination about eleven o'clock at night. If you had lost count of the days, 
what would you conclude to be the approximate date ? 

8. On April 1, 1895, the moon's R.A. was approximately 23 hours 48 minutes. 
Give roughly its appearance, time of rising, and transit on that date. 

9. If the sun and the Great Bear were together visible throughout one, and 
only one, night in the year at some locality on the Greenwich meridian, how 
could you determine its distance from London, if you also remembered that 
the earth's diameter is very nearly 8,000 miles, and the latitude of London is 
very nearly 51? 

10. If you observed that the sun's noon shadow vanished on one day of the 
year and pointed south on every other, how many miles would you be from the 
North Pole? 

11. On January 1st the sun reached its highest point in the heavens when the 
B.B.C. programme indicated 12.17 p.m. It was then 16 above the southern 
horizon. In what part of England did this happen? 

12. The approximate R.A. and declination of Betelgeuse are respectively 
5 hours 60 minutes and 7 N, If your bedroom faces east, and you retire 
regularly at 11 p.m., about what time of the year will you see Betelgeuse rising 
when you get into bed? 

13. On April 13, 1937, the shortest shadow of a pole was exactly equal to its 
height, and pointed north. This happened when the radio programme timed 
for 12.10 p.m. began. In what county were these observations made? 

14. With a home-made astrolabe the following observations were made in 
Penzance (Lat. 50 N., Long. 5J W.) on February 8th: 

Least Zenith Distance Greenwich Mean Time 
Betelgeuse 42 S. 9.9 p.m. 

Rigel 58 S. 8.24 p.m. 

Sirius 674 S. 10.0 p.m. 

Find the declination and R.A. of each star, and compare your results with the 
table in Whitaker's Almanack. 

15. By aid of a figure show that if the hour angle (/*) of a star is the angle 
through which it has turned since it crossed the meridian (or, if the sign is 
minus, the angle through which it must turn to reach the meridian), 

R.A. of star (in hours) Sun's R.A. (in hours) (hour angle in 
degrees -~ 15) + local time (in hours). 



CHAPTER III 

SPECTACLES AND SATELLITES 

The Trail of the Telescope 

AMONG writers who are not familiar with the history of science, it has been 
the fashion to speak of the great intellectual awakening of the fifteenth and 
sixteenth centuries as if it were closely connected with the artistic Renaissance 
of Italy under Byzantine influence. The truth is that the advance of science 
owed very little to the influx of classical models and classical texts from the 
Eastern Empire. The fruits of Alexandrian science were harvested by the 
Arab conquerors of Spain; and the diffusion of the Arabic learning into 
northern Europe was largely due to the influence of Jewish physicians who 
founded the medieval schools of medicine, and to the development of scientific 
navigation before the tradition of the Moorish universities had been finally 
extinguished. Two features of Catholic tradition and organization forced 
the universities of western Christendom to open their doors to the Moorish 
learning. One, which will be discussed in Chapter XVI, was the humani- 
tarian ideology which prompted the monastic orders to found hospitals and 
seek the assistance of experienced physicians. The other, which encouraged 
monks like Gregory of Cremona or Adelard of Bath to visit the universities 
of Spain during the Moorish occupation, was the social function of the 
priesthood as custodians of the calendar. In conformity with their role as 
timekeepers, the Augustinian teaching had endorsed astronomy as a proper 
discipline of Christian education. So although the patristic influence was 
mainly hostile to pagan science, clerical education was not completely in- 
accessible to influence from the non-Christian world. 

A substantial link between Moorish science and the medieval universities 
of Christendom already existed when the growth of mercantile navigation 
renewed the impetus to astronomical discovery. About 1420 Henry, then 
Crown Prince of Portugal, built an observatory on the headland of Sagres, 
one of the promontories which terminate at Cape St. Vincent, the extreme 
south-west point of Europe. There he set up a school of seamanship under 
one Master Jacome from Majorca, and for forty years devoted himself to 
cosmographical studies while equipping and organizing expeditions which 
won for him the title of Henry the Navigator. For the preparation of maps, 
nautical tables, and instruments, he enlisted Arab cartographers and Jewish 
astronomers, employing them to instruct his captains and assist in piloting 
his vessels. Peter Nunes declares that the Prince's master mariners were well 
equipped with instruments and those rules of astronomy and geometry 
"which all map-makers should know." The development of astronomy once 
more became part of the everyday life of mankind, and the new impetus 
it received from the growth of maritime commerce was reinforced by the 
introduction of two new technical inventions which emerged from the 
world's everyday work in a different social context. 



132 



Science for the Citizen 



One of these was the invention of spectacles. Although devices of one kind 
or another for magnifying objects are of considerable antiquity, there does 
not seem to have been any general use of them in everyday life till the close 
of the Middle Ages. The Moorish savant Al Hazen appears to have observed 
the magnifying power of the segment of a glass sphere (Opticae thesaurus, 
vii, 48); and Roger Bacon explained in his Opus Majus (1266) how to magnify 
writing by placing the segment of a sphere of glass on the book with its 
plane side downwards. One of the earliest examples of the application of 
the principle is provided by a portrait of Cardinal Ugone in a church fresco 




FIG. 67A 

Ptolemy's Map of Ireland showing what the Great Navigations owed to Alexandrian 
Cartography. Note the lines of latitude and longitude. 

at Treviso. This was painted in 1352. It shows two mounted lenses with their 
handles riveted together and fixed before the eyes. In DobelFs book, Anthony 
Leeuwenhoek and His Little Animals, the actual inventor of spectacles is 
said to be a Florentine about A.D. 1300. The invention was popularized by 
the monks, notably through the private labours of de Spina of Pisa for the help 
of "poor blind men." The compound microscope appears to have been in- 
vented at the same time as the telescope, possibly, according to Professor 
Wolf, a little earlier. It is mentioned in a letter dated 1625 by Giovanni 
Fabri, who uses the word. Pictures made with the help of the microscope by 
Stelluti were published (1630) in an Italian edition by Persius. Between 
1650 and 1670 microscopic observations were published by Borel, Power, 
Hooke, and Leeuwenhoek. With the growth of literacy, the universal ten- 
dency to long sight as age advances found expression in a new social demand. 
The introduction of spectacles involved no theoretical discovery about 



Spectacles and Satellites 133 

phenomena with which Alexandrian and Arab astronomers were not fully 
conversant (see p. 141). It is therefore more reasonable to suppose that 
introduction of paper, the invention of printing, and the use of books in the 
fifteenth century stimulated the demand for eye glasses. The trade increased 
during the sixteenth century, especially in Italy and in south Germany. By 
1600 opticians were to be found in most of the larger towns on the continent. 

Two inventions which are signposts in the history of science, one in 
physics, one in biology, came as quite fortuitous by-products of the new 
industry. The name telescope was first adopted by Galileo in 1612. The 
first one was invented in Holland about 1608. The credit has been attributed 
variously to Hans Lippershey and Zacharias Jansen, spectacle makers in 
Middleburg, and to James Metius of Alkmaar (brother of Adrian Metius, 
the mathematician). On October 2, 1608, the assembly of the States-General 
at the Hague considered a petition of Hans Lippershey, inventor of an 
instrument for seeing at a distance. On October 9th, 900 florins were voted 
for it. On December 15th they examined, and voted 900 florins for, a binocular 
instrument made by Lippershey. Descartes attributed the invention of the 
telescope to Metius, who presented a petition later. The story goes that 
the discovery was made by holding two pairs of spectacles some distance 
apart and noticing that a neighbouring spire was brought nearer to view. 
In his Nuncius Siderius Galileo states that when he was in Venice about 
May 1609 he heard that a perspective instrument for making objects appear 
nearer and larger had been invented. Returning to Padua, he made his first 
telescope by fixing a convex lens in one end of a leaden tube and a concave 
lens in the other end. Then he made a better one, went to Venice, and pre- 
sented the instrument to the Doge Leonardo Donato. His first telescope 
magnified 3 diameters. He soon made others which magnified 8 diameters, 
and finally one that magnified 33 diameters. Kepler devised an alternative 
form using a convex eyepiece. 

The three years which followed the patents of Lippershey and Jansen 
were eventful. Kepler's account of the motion of Mars appeared in 1609. 
His telescope was constructed in 1611. Eight years later he was able to 
announce his complete vindication of the fundamental doctrine of Copernicus 
and his epoch-making laws of the solar system. Meanwhile Galileo had 
observed the motion of the sun's spots and had seen the moons of 
Jupiter. Galileo's discovery was partly important because it deprived the 
geocentric view of the universe of the inherent plausibility it enjoyed before 
people realized that there were other worlds with satellites circling about 
them. The Inquisition rightly judged the psychological effect of the new 
realization that our own small world is not a unique one. How the tract on 
the moons of Jupiter became the field of one of the most decisive battles 
between science and priestly superstition is a story too familiar to merit 
repetition. 

The telescope has a threefold significance for the age of the Great Naviga- 
tions. The determination of longitude for westerly sailing had become a 
technical issue of cardinal importance, and on this account astronomy 
retained its place as the queen of the sciences till the end of the eighteenth 
century. At a time when the only method of determining longitude was based 



134 



Science for the Citizen 



on the use of celestial signals (eclipses and conjunctions), celestial signals were 
events of vital significance for the world's work; and the discovery of Jupiter's 
moons brought a new battery of celestial signals to the aid of seafaring and 
scientific geography (see p. 315, Chapter VI). More directly the telescope 
was of value to the mariner as a "spy glass." A less obvious use is related 
to one of the pivotal inventions in the history of mankind." The age of the 
Great Navigations was a period of revolutionary and imperialist wars in which 
success depended on exploiting the new technique of artillery. The demands 
of marksmanship called for accurate and immediate devices for surveying 
(see Fig. 149) and sighting distant objects. Galileo was not slow to recognize 
the possibilities of the telescope for military use. Indeed, he offered his 




, , 



\ 

FIG. 68 

As the edges of a beam seen emerging from a slit are always straight, we can look 
upon a shaft of light as made up of bundles of straight shafts or rays, which Alexandrian 
astronomers represented by geometrical lines or "rays." Light rays falling upon a 
surface which is opaque are scattered in all directions. When we look at anything 
only a few rays are able to enter the eyeball. 

invention consecutively to the Catholic Emperor and to the opposing 
Protestant democracy in letters adapted to the convictions of both parties.* 
The invention of the telescope is an instructive episode, if we are disposed 
to interpret the progress of human knowledge as an unbroken chain of 
theoretical principles due to the exercise of mere curiosity by a few excep- 
tionally gifted individuals. The fundamental laws of magnification were 
familiar to the Alexandrians. Euclid composed a work on the geometrical 
principles of reflection, and Archimedes is credited with constructing convex 
mirrors for use as burning-glasses. Ptolemy investigated refraction, i.e. the 
bending of a beam of light in passing from one medium to another, and 
actually discussed the effect of atmospheric refraction in distorting the 
apparent position of objects on the horizon. Al Hazen, an Arab physician 
who lived about A.D. 1000, gave a correct account of the structure of 
the eye, contested Plato's idealism, which made it the source of illumina- 

* I owe this information to Dr. J. D. Bernal, who found it among a collection 
of Galileo's literary remains. 



Spectacles and Satellites 135 

tion, and appears to have recognized it as what we should now call a camera. 
There was no further development of the subject till the invention of the 
telescope. The design of better telescopes immediately created two needs. 
The need for more definite theoretical principles of guidance in attaining 
high magnification led inevitably to a more precise statement of the law 
of refraction by Kepler, Snell (1621), and Descartes (1637). The need to 
eliminate the coloured fringe which blurs the outline of an image obtained 
with simple lenses led Newton to the study of the spectrum. 

A number of circumstances conspired to encourage the experimental study 
of light at an early stage in human knowledge. Out of the quagmire of Greek 
idealism there had emerged one very pertinent issue. How can we tell when 
our senses deceive us? From the very dawn of civilization possibly earlier 
than any cities man had become accustomed to the use of the mirror as an 
aid to adornment. Earlier still he had puzzled about reflections seen in 
pools. Things do not always seem to be where they are or to be the right 
way up. Early interest in such illusions is illustrated by the trick of the 
disappearing coin, which was correctly interpreted by Ptolemy as an effect 
due to the bending outwards of light when it passes from water into air, as 
shown in Fig. 69. Optical illusions, which possibly enjoyed a place in the 
priestly magic, assumed a special practical importance in an age when 
people were intensely interested in the heavenly bodies and were beginning 
to have a scientific knowledge of them. 

There are several very striking optical illusions in the everyday life of people 
who watch for the rising and setting of the sun and moon. When near the 
horizon the sun and moon appear to be much larger than when seen high 
in the heavens. When seen rising behind a hill they seem to move much 
more rapidly than they do when they are above us. Measurement of the sun's 
or moon's angular diameter by an instrument, or of their bearings when 
rising and setting do not confirm our first impressions. Al Hazen was among 
the first to realize what we know to be a correct explanation of the deception. 
Our estimates of sizes and distances near the horizon are peculiar because we 
adopt comparatively near terrestrial objects as our standard of comparison. 
In learning to use the astrolabe or quadrant, men were taking the first eventful 
step of manufacturing social sense organs which do not deceive us in arranging 
the conduct of everyday life. One important fact in connexion with the 
early development of what we now call geometrical optics is that the kind 
of mathematics which developed in connexion with astronomy is specially 
useful for measuring the elementary properties of light. 

The Narcissus myth still pursues us in everyday life from the moment 
we get up in the morning and shave or powder before a mirror. It no 
longer provokes among sensible people the mystification which led early 
philosophers to devise the false antithesis of appearance and reality. Today 
the experience of photography is enough to dispose of Plato's belief that the 
eye sends forth light. Appearance is part of the reality which includes a correct 
account of how our sense organs do their work. We know enough about 
the sense organs to see clearly what early science could only grope after in 
the dark. Vision is not merely a static copy of the changing world. We have 
to learn to use our eyes as we have to learn to use any other instrument. 



136 



Science for the Citizen 



Our estimates of distance, direction and position do not merely depend on 
what the retina of the eye tells us. They involve complex movements of the 
muscles that move the lens of the eye, of the muscles that change its 
curvature and of the muscles that wield the eye itself in its socket, together 
with movements of the muscles of the neck and limbs and all the signals 
which these muscles send to the brain when they move. We learn to co- 

B 





FIG. 69 

The disappearing coin trick is of very great antiquity and provided the Alexandrian 
astronomers with a correct explanation of certain astronomical phenomena, such as 
why the apparent time of setting of the sun is a little later and its apparent rising 
a little earlier than expected from calculations as given in Chapter IV, p. 197. When 
the vessel is empty the eye cannot see deeper than the level of A, and the coin can 
only be seen if the head is raised till the eye is at B. If water is poured into the vessel 
the coin at the bottom becomes visible. When the sun is just below the horizon in 
the geometrical sense sunbeams passing from rarefied space into the denser atmo- 
sphere of the earth are bent downwards, so that the sun remains visible for a short 
while after it is actually below the level of the horizon, just as the coin is visible in 
the lower figure though below A. 

ordinate our bodily movements with those of our eye muscles and with the 
pattern of light on the retina from the common experience of everyday life. 
Part of our everyday experience is that a beam of light has a straight 
edge, or to use a customary figure of speech, "light travels in straight 
lines." We learn to put things "in line." The delicate adjustment which 
makes us able to grasp a thing by seeing it, or to direct our gaze to the thing 



Spectacles and Satellites 



137 



we touch, is easily upset. By a simple device shown in Fig. 70 we can 
direct our eye movements so that we infer a set of lines which are really parallel 
to look as if they converge or diverge. By using mirrors, or looking at an 
object immersed in a different medium from that in which we usually see it, we 
can make a beam of light change its course from an uninterrupted straight 
line. So our eyes lead us to make judgments that do not agree with those 
which we are led to make when we also use our organs of touch. In such 
situations we call what we see an image of the thing or object as we see it when 
the testimony of all our sense organs agrees. Real images which can be 



FIG. 



.v -/ 
70. Two OPTICAL ILLUSIONS 



Judgments of vision do not merely depend on the static picture focussed on the 
retina at a given instant of time. They involve active movements of the eye muscles. 
If these movements are biassed our geometrical judgments go awry. Thus the two 
lines AB and CD in the upper half of the figure are the same size, and the vertical 
lines in the lower half are all parallel. You can see that they are, if you half shut your 
eyes. 

caught on a screen are distinguished from virtual images which cannot. 
The image on the ground glass of a camera or the image of the sun 
focussed with a burning-glass are familiar examples of the former. Re- 
flections in flat mirrors are familiar examples of virtual images. 

The elementary principle that "light travels in straight lines" was implicit 
in all the astronomical lore of the ancient world. So was one of its conse- 
quences on which, as we shall see later on, depends our knowledge that the 
earth's path around the sun is not a perfect circle, as Plato believed. The 
apparent size of a body depends on the angle which light reflected from its 
outermost edges makes with the eye of the observer. We call this angle the 
angular diameter. Fig. 71 shows the same spherical object drawn at two 



138 Science for the Citizen 

different distances from the eye. When it is nearer, the angular diameter is 
larger than when it is far away. The sine of half the angular diameter is 
the ratio of its radius to its distance from the observer. When the angle is very 
small the ratio of two radii is not appreciably different from the ratio of the 
two angular diameters (Fig. 71, legend). Hence the apparent length of the 
radius of a circular or spherical body seen in different situations is inversely 
proportional to the distance. When we magnify a body by a mirror or a lens 
we change the path of the light coming from it so that the angle between the 
beams or "rays" converging on the eye is increased. The magnification of a 
very distant object is thus the ratio of the angular diameter of this seemingly 





FIG. 71. APPARENT SIZE AND DISTANCE 

The same object seen near and far away. The angles a and b are called its angular 
diameters as seen in the two situations. If r is the actual radius and d the distance^ 
sin la = r -~ d, and sin %b = r ~ D. When very small angles are measured in circular 
measure (i.e. ratio of the arc to the radius of the circle of whose circumference it is 

part), <z - sin \a = PQ ~ d and \l = PQ ~ D (see Fig. 213). Hence r = -? 

D a 

nearer image to the angular diameter of the object seen without the 
instrument. 

Modern telescopes use curved mirrors as well as lenses for magnification. 
Two experimental laws are sufficient to deduce the essentials for designing 
mirrors and lenses so as to produce a known degree of magnification, whether 
in constructing a telescope, prescribing the right spectacles or for the variety 
of other uses to which magnifying devices are now put in everyday life. The 
fundamental law of reflection is that when a shaft of light strikes a flat 
polished surface it is bent outwards in the same plane, making equal angles 
with an imaginary perpendicular drawn where it hits the mirror, as seen in 
Fig. 72. That this is approximately true is easily seen when a bright shaft 
of light falls through dusty air on a polished surface. It can be tested more 
precisely by the use of pins and a mirror placed edgewise on a sheet of paper 
as explained in Fig. 73. The Alexancjrian astronomers were familiar with the 



Spectacles and Satellites 



139 




TVfJRROR, 

FIG. 72 

The fundamental law of reflection is that a beam of light incident on a flat surface is 
reflected in the same plane so that the angle of incidence (a) is equal to the angle ol 
reflection (b\ 




FIG. 73 



A simple way of testing the law of reflection is to fix two pins, A and B, in front of a 
mirror placed edgewise on a piece of paper. By moving the head from side to side,a 
point is found at which the "virtual" images, A x and B! are, in line. Two pins, C and D, 
are now placed in line with A x and E,. Lines can then be drawn through the pin 
holes and the angles i and r, formed with the perpendicular are found to be equal. 



140 



Science for the Citizen 



application of the law to the formation of magnified images by curved 
surfaces. The second principle which is necessary for understanding how 
magnification is produced by lenses is the law of refraction or the bending 
of a beam of light when it passes from one transparent medium into another. 
This bending is beautifully seen when a shaft of light passes through an 
aquarium tank with glass sides as in Fig. 74. 



uiiiiiruiiiiiiiiiiiiiiiiiiiitfiiiiiiimiiiiiiHliHiiiiiiiiiiHim 




FIG. 74 

The refraction of light inwards when a beam of light passes into an aquarium tank 
filled with water and outwards when it passes into the air again. Passing^from air to 
water the incident angle i 45 in the figure, and the angle of refraction" (r) is given 
by Snell's law, viz. sin i = sin r x R, i.e. 

sin r = sin 45 -f R. 

Very accurate determination of R for air and water gives the value 1 3 or 4/3. From 
the tables sin 45 = -7071. 



sinr =-- -7071 -f- =- 

o 



5303. 



Tables give sin 32 = 5299. So r is very nearly 32. In using the formula remember that 
R is 4/3 for light pas sing from air to water, i.e. if R is 4 /3, i is always the angle which the 
ray makes with the vertical in air and r is always the angle which it makes with the 
vertical in water, irrespective of th actual direction of the beam. If, however, we 
apply the terms "angle of incidence" and "angle of refraction" literally, then R is 3/4 
for light passing from water to air. The first convention saves us^the trouble of 
recording two values of R for each pair of media. 

The technology of the telescope was not what first directed attention to 
it. In contradistinction to the physiological distortion of objects seen near 
the horizon, when viewed with the unaided eye, another class of optical 
puzzles arises in the study of celestial phenomena, especially in connexion 
with a practical problem on which the collection of fines from motorists 
depends today. As you know, lighting-up time is fixed when "civil twilight" 



Spectacles and Satellites 141 

ends, and this rests on calculation of the times of rising and setting of the 
sun. From geometrical principles ou dined on p. 99 and explained more fully 
in the next chapter, the times of rising and setting of a heavenly body are 
calculable from their position (R.A. and Declination) in the celestial sphere. 
Such calculations do not exactly correspond to the observed times of visi- 
bility at the horizon level unless a small correction is made for an illusion 
which depends on the fact that light is bent inwards on passing from empty 
space into the earth's atmosphere. The amount of this bending depends on 
the angle from which the object is seen, and has nothing to do with the 
imperfection of the eye as a physical instrument. So observations on the 
direction of the sun, moon, or stars, viewed near the zenith, do not accurately 
correspond to observations made near the horizon. To put the matter in 
another way, the sun is not geometrically in line with the horizon edge at the 
moment when it is seen setting. It is already a little below it, as the penny 
is a little below the line joining the edge of the basin to the eye in the dis- 
appearing coin trick. 

The astronomers of Alexandria were fully conversant with the existence of 
refraction, and realized that it provided the explanation of the disappearing 
coin illusion, and why bodies immersed in water seem to be nearer to the 
surface than we find them to be when we try to grasp them. The attempt 
to find a correction for "atmospheric refraction" led Ptolemy of Alexandria 
to make the first extensive physical experiments in which numerical measure- 
ments were recorded to discover the amount of bending. In his experiments 
he studied the passage of light from air to water, from air to glass, glass to 
water, and vice versa. For water and air, he used a piece of apparatus essen- 
tially like that shown in Fig. 75. Although Ptolemy obtained numerical 
data good enough for making an empirical correction for the bending of 
light in passing from one medium to another, his results were not accurate 
enough to display the simple geometrical rule which more careful observation 
might have disclosed. Though atmospheric refraction does not affect observa- 
tions on the time of transit, it does affect observation of the exact time when a 
heavenly body is seen in a given position east or west of the meridian. 
When the invention of the telescope revived interest in optical phenomena, 
interest in atmospheric refraction was renewed by the attempt to make 
accurate determinations of the moon's R.A. and the positions of the planets 
by observations when they are not on the meridian with the help of 
methods explained in the next chapter. 



THE NATURE OF A SCIENTIFIC LAW 

There is much misunderstanding concerning the meaning of a scientific 
law. So it may be well at this stage to illustrate the meaning, scope, and 
method of arriving at the law of refraction by using the first recorded body 
of numerical data in the history of experimental science. By referring to 
Figs. 74 and 75 you will see what is meant by the angle of the incident ray 
(i) and the angle of the refracted ray (r) when light passes from air into water. 
The terms are relative to the direction from which we are supposed to view 
the source of light. If light passes from water to air it is bent outwards and 



142 Science for the Citizen 

the incident angle is the smaller of the two. If light passes from air to water 
it is bent inwards and the angle of refraction is the smaller. The figures obtained 
in one of Ptolemy's experiments* are given in the two left-hand columns of 
the succeeding table. 

Angle of Refraction 

Angle of Calculated by 

Incidence (t) Observed (r) Proportional Parts Percentage Error 



10 8 7| 3-1 

20 15J 15J 1-7 

30 22| 22J 1-1 

40 29 28 J 0-9 

50 35 34f 0-7 

60 40| 40J 0-6 

70 45 J 45J 0-6 

80 50 

Suppose we are satisfied that every one of the figures in this table is 
correct. That is to say, repeated observations agree. We can use the figures in 
one of three ways to calculate how much a ray entering water will be bent. 
The first is arithmetical, and is called the method, of proportional parts. 
To find the angle of refraction when the incident angle is 22 we should 
refer to the observed values tabulated as above, finding that the two nearest 
figures for the incident ray are 20 and 30. An incident ray of 22 is obtained 
by adding A or i of the interval (10) between 20 and 30 to the former 
figure. The corresponding angles of the refracted ray are 15 J and 22|% the 
interval being 7. Adding \ : x 7 to 15i we get 16-9. To see how far this 
procedure is satisfactory we may compare the values of the refracted angle 
in Ptolemy's experiment calculated on the same assumption with the values 
he actually obtained. Thus 10 is half the interval between and 20 and 
the corresponding angles of refraction are and 15J. Half the interval 
between and 15 J is 7| instead of 8 as the experiment gives. We then get 
the values given in the third column. The fourth column, showing the 
percentage error of the calculated as compared with the observed value, 
indicates that the greatest discrepancy is 3*1 per cent, the average being 
1 -2 per cent. For incident angles between and 70 we might reasonably 
conclude that such calculations would not generally lead us to make an error 
much larger than 3 per cent, and rarely if at all more than 4 per cent. A 
second way of using the information provided in the table would be to plot 
the values on squared paper, draw a smooth curve as nearly as possible 
through them, and read off the values which we wish to find out. Either of 
these methods can be made more accurate by including as many observations 
as we care to make in our table or graph of recorded results, and both are 
open to the objection that we have to refer to the original data whenever 
we wish to make a calculation. 

The third method of using the data is to find some simple expression 
from which all the observed values can be inferred with a fairly high degree 

* As given in the translation cited by Brunet (Histoire des Sciences: Antiquite). 



Spectacles and Satellites 



143 



of precision. Once we have found such an expression and ascertained how 
big an error can arise in using it, we have no need to refer again to the original 
data. Such an expression is the sort of condensed statement of observed 
truths known as a scientific law. Ptolemy did not himself succeed in detecting 
any simple rule of this kind connecting the numbers he obtained. The truth 
is that his observations on bending of the smaller and larger angles (10% 20, 
and 80) were not very accurately made. With a device like the one in Fig. 75 




FIG. 75 

A simple apparatus for finding the law of refraction^ which can be made with a fret- 
saw from three-ply wood. At A, B, C nails project. When one bar is held in any posi- 
tion the other is rotated till all three projecting nails appear to be in line. From the 
graduated scale of degrees, the "angle of incidence/' i.e. the angle light makes with the 
vertical in air, given by the bar BC, can be compared with the "angle of refraction," 
i.e. the angle light makes with the vertical in water, made by the bar AB. 

In this instance the source of light is the light reflected from the bottom pin A, 
and a ray following the path AB is refracted outwards in air along the path BC to the 
observer's eye. It is immaterial which we call the angle of incidence and the angle of 
refraction as long as we always keep one term for the angle of the light ray in air and 
the other for the angle of the light ray in water. Literally speaking the latter is the 
incident ray in this case, but the law is usually stated as for light travelling in the 
opposite direction ; in which case the terms have the meaning given above. 



a schoolboy can get data which agree more closely with careful observations 
carried out with good instruments. 

In seeking for a simple formula connecting two sets of numbers such as 
those in the first left-hand column (/) of the preceding table and the corre- 
sponding figures in the next column (r) the investigator may be guided 
by the shape of a curve based on his observations, or by analogy between 
the process that he is studying and some other process for which he already 
knows a satisfactory law. He may have to try several ways of stating the 



144 



Science for the Citizen 



connexion before he finds one that is satisfactory. Kepler proposed the 
formula: 

Kr cos r 



K cos r - (K ~ 1) 
Snell and Descartes gave the law as : 

sin i = R sin r 

In these two formula K and R are "constants," i.e. fixed numbers which 
are the same for any pair of media, e.g. air to water, water to air. For different 
pairs of media different values of R or K must be used. These can be tabu- 





3 Ray at grazing in 




FIG. 76. LIMITS OF REFRACTION 
Four rays from the bottom and sides of a vessel converging at the same point P. 

1 (angle of refraction zero) is not bent, when passing out into air. 

2 (angle of refraction less than critical angle, C) bent towards the surface on 
emerging into air. 

3. (angle of refraction equal to the critical angle) emerging ray just grazes the 
surface. 

4. (angle of refraction greater than the critical angle) does not emerge but is reflected 
downwards. 

lated in a short list of refractive indices. This dispenses with the need for 
constant reference to all the original experiments carried out to ascertain 
the law.* 

Having found a formula which seems to connect two sets of measurements 
in a physical experiment the first thing is to ascertain how well it fits them. 
As Ptolemy's figures for smaller and larger angles do not agree with careful 

* In using them, of course, we must remember, that in passing from water to 
air instead of air to water, f will be the angle which the beam makes with the vertical 
in water and r will be the angle it makes with the vertical in air. The values of R are 
usually tabulated for light passing from the lighter to the heavier medium. Otherwise 
the reciprocal of the tabulated value is used. See also p. 148. 



Spectacles and Satellites 145 

observations which any competent person can make, we may shorten his 
table to illustrate how this is done. In Kepler's and Snell's laws K and R are 
constants for a given pair of media. How far this statement is true is seen 
from the following table which you will find it instructive to check. For 
instance, on looking up sin 20 in tables of sines it is found to be 0-342 and 
sin lf>! = 0-267, so that by Snell's formula R = 0-342 ~ 0-267 = 1 -28. 

Kepler's formula Snell's formula 

K ^ * R _ sin? 

i r i + (cosr)(r z) sin r 

20 15J 1-28 1-28 

30 22i 1-30 1-31 

40 29~ 1-32 1-33 

50 35 1-33 1-34 

60 40 1-33 1-33 

70 45| 1-33 1-32 

Having done this to find a representative value of the physical constant 
characteristic of the substances which we are using, we have next to test the 
accuracy of the law we propose to use. Taking the mean of the values in the 
last table as representative, K = 1 32 in Kepler's law and R = 1-32 for 
SnelPs law applied to water and air, we may check the accuracy of each law 
by comparing values of r calculated by the formula with values actually 
observed. For instance, by Snell's law, sin r = sin i ~ R. If i is 70, sin i 
= 0-9397 and sin r = 0-9397 ~ 1-32 = 0-7119. The tables give 0-7119 
as sin 45 38, so the calculated value of r, (45 38), differs from the observed 
(45i) by 1 2, and an error of 1 2 on 45 is 3 in a hundred or less than 
5 per cent. Making a table of the percentage errors obtained in calculating 
values of r from values of i by the two laws we get the following : 



I 
Observed 


i Calculated 
from r by 
Kepler's Law 


Percentage 
Error 


r 
Observed 


r Calculated 
from i by 
Snell's Law 


Percentage 
Error 


20 


20-71 


3-6 


15* 


15-02 


3-1 


30 


30-51 


1-7 


22 i 


22-27 


1-0 


40 


40-12 


0-3 


29 


29-15 


0-5 


50 


49-71 


0-6 


35 


35-47 


1-3 


60 


59-46 


0-9 


40i 


41-00 


1-2 


70 


69-56 


0-6 


451 


45-38 


0-3 



Mean percentage error 1 3 Mean percentage error 1 2 

Taking the facts as they stand, i.e. assuming that Ptolemy's data are as 
accurate as we can make them by using the best instruments and the 
most careful record of our readings, we see that between 20 and 70 
we can calculate the angle of refraction correct within 3 per cent, or 
within half a degree, by using Snell's law. Clearly, also, there is not 
much to choose between the accuracy of the two laws with only this 
information to guide us. To decide in favour of one or the other we may 
next compare the results of applying both laws to other media, e.g. water 
and glass, or glass and air, or, using the same apparatus as before, air and 
paraffin, air and alcohol, air and glycerine, or water and olive oil. It might 



146 Science for the Citizen 

happen that the average error was much larger for some media than for 
others, using one law, while the other gave results of equal consistency. We 
should then have a reason for deciding in favour of the latter. It might 
happen that the differences between observed results and values calculated 
by either law were never greater than the accuracy with which we could 
make the observations. For instance, it might happen that repeated observa- 
tions would not agree within half a degree, and that both sets of calculated 
values would not differ from the observed average values by more than half a 
degree. Either law would then be a safe guide to correct conduct, and our 



Setting \49 

Sizn 




FIG. 77 

One queer thing about the fish's heaven is not mentioned in Rupert Brooke's poem. 
The sun sets in the heaven of the fish at an angle of 49 instead of 90 from the zenith 
and the whole terrestrial world which is visible to the fish is compressed in a cone 
of which the apex is 2 x 49. When the sun is just dipping on the horizon its rays 
enter the water at nearly 90 from the vertical, being then bent to 49 from the vertical. 
The result is that the sun appears to be at P. 

only reason for preferring one to the other would be that using it involved 
less effort. 

In this case SnelPs law is the simpler of the two. That is to say, we can 
calculate more quickly with it. However, it should not be regarded as the true 
law of refraction merely because it is simple. It is more true if it yields 
results which agree more closely with observed facts. Even if it were less 
accurate than Kepler's law we might still use it in certain situations. If 
you only want to know the angle of refraction within a degree, a law which 
tells you how to calculate it with an error of half a degree is just as good as 
a law which tells you how to calculate it within a hundredth of a degree. So 
if the first happens to involve less effort spent in calculation you will naturally 
prefer it to the second. In testing a law there is another important thing 
to remember. Our table shows how the law helps us to calculate the angle 
of refraction for incident angles between 20 and 70. Would it be true from 



Spectacles and Satellites 



147 



to 20 or from 70 to 90? The data do not tell us. But two simple experi- 
ments suffice to show that SnelTs law holds good at the extreme limits. A 
beam of light passing vertically downwards is not displaced to the right or 
left, i.e. when i = 5 r = 0. SnelTs law gives sin r = R = 0, i.e. r = 0. 
If a beam of light just grazes the surface of the water the angle of incidence 

is 90. Snell's law shows that sin r = sin 90 -=- R = ~. Taking the mean 

JK 

given above, sin r = 1 -r 1-32 = 0-7576. The tables tell us that 0-7576 
is the sine of 49J or 49 to the nearest degree. This means that a beam 
of light coming from air into water cannot be bent in so that the angle of 



'cct 





FIG. 78 

Two prisms (with angles 90, 45, 45) arranged as in the figure constitute the essential 
parts of a periscope. Parallel rays from the object enter the first face at right angles 
and are not refracted. The unrefracted ray strikes the slanting surface at 45 from the 
vertical. This is greater than the critical angle for air and glass so the ray does not 
pass out into the air. It is reflected downwards. 

refraction is more than 49 (i.e. bent downwards 90 49 = 41) as shown 
in Fig. 76. Conversely if a beam of light in water is inclined to the vertical 
at 49, (r = 49), it will be bent outwards so much that it will just graze the 
surface as it passes into air, and if the "angle of refraction" in water is greater 
than 49 a beam of light will not pass out into air at all. It will be totally 
reflected backwards at the surface (Fig. 76). This is called the critical angle, 
and can be determined very accurately in agreement with Snell's law. For 
air into flint glass the value of R is 1 -65 and (1 ~ 1 -65) = 0-606 = sin 37 
(nearly). So the critical angle for flint glass is 37. Thus a prism with angles 
45, 45 and 90 can be used as a perfect mirror if placed as in Fig. 78. 
Prisms are used as reflectors in binoculars and periscopes. 



148 Science for the Citizen 

The main points to grasp about a scientific law based on measurements is 
that it is : (a) a condensed statement of observed facts to use as a guide for 
practical conduct, (b) the most economical way of getting a result as accu- 
rately as we need it. So to say that a law has been superseded does not mean 
that it is wrong in an absolute sense. It is wrong to go on using a law when 
it is no longer sufficiently accurate for our needs, especially when we can 
summarize what is known in a way which is sufficiently precise for use. 
Right and wrong in the domain of scientific laws is partly judged in its his- 
torical context, as believers judge the family life of the patriarchs. There is 
also a third sense in which a scientific law is not absolute. At certain limits it 
breaks down. In the quaint language used by scientific workers of the Soviet 
Union this is expressed by saying that "quantity passes into quality." So a 
third point about using a law is that we are not entitled to apply it beyond 
the limits of the original data until we have ascertained within what limits 
it is true. When the incident angle of a beam of light passing from water into 
air is greater than 49 the law of refraction ceases to apply and another law, 
the law of reflection, has to be used. 

To use Snell's law we need to know R for the two substances which we have 
to deal with, e.g. water and air, glass and water, etc. For any pair of them, 
R can be found by experiment once for all. The values which have been 
found by careful experiment are tabulated in books of "physical constants." 
As given in them, it is usually understood that the light passes from the 
lighter medium (e.g. air) to the heavier medium (e.g. glass). So, for purposes 
of calculation, the angle of incidence is the angle of the beam in 
the lighter medium, and the angle of refraction is the angle of the beam in 
the heavier medium, irrespective of the actual direction of the beam. This 
convention saves us the trouble of recording two values of R for each pair 
of media, e.g. R = f for air to water, and R = f for water to air (Fig. 74). 

The tables usually give only refractive indices of transparent substances with 
reference to air. If you know the figures for R between air and two other sub- 
stances, you can calculate the value for R when light passes from one of them 
to the other by a simple rule which is easy to expose in a scale diagram like 
Fig. 80. The rule is 

R (B to C) = R (A to C) ~ R (A to B) 

Thus for air to flint glass, R is given in the tables as 1 * 65. For air to water it is 
1 33. Hence for water to flint glass it is 1 - 65 ~ 1 - 33 = 1 -24. 

ACCURATE DETERMINATION OF THE REFRACTIVE INDEX 

From what has been said, it is clear that we need better data than the ones 
which Ptolemy recorded to decide between the merits of Kepler's formula 
and Snell's law. With properly constructed instruments it can be shown that 
the latter is more accurate. The value of R (or the refractive index) of sub- 
stances does not vary more than 0-01 per cent when calculated according to 
Snell's law. 

One method of determining R for different substances is illustrated in 
Fig. 79. Another is shown in Figs. 80 and 81. When light passes through 
transparent substance the emergent beam is parallel to the incident beam, if 



Spectacles and Satellites 



149 




Duplicate 




apparent 



FIG. 79 

One way of testing SnelTs law or of finding the Refractive Index (R) for a liquid is 
to compare the actual depth of an object immersed in a tall cy Under of the liquid 
with its apparent depth judged by holding a duplicate outside the cylinder in such a 
position that both the immersed object and its duplicate appear at the same level when 
the head is moved from side to side. If SnelTs law is true: 



R 



real depth 
apparent depth 



In the diagram O is an object immersed. The ray OB makes the angle r with the 
vertical and is bent outwards making the angle i with the vertical on passing into air, 
so that the object appears as if it were at I. By parallels 

i=Z.AIB and r = Z.AOB 

A T> AT? 

sin AIB = -== and sin AOB = -^=z 



sin* ^ sin AIB ^ OB 
sin r~ sin AOB IB 

When we are looking downwards the angles i and r are very small, so that 

BO __ AO real depth 

BI AI apparent depth 

For water R = 1 J so that the apparent depth is three quarters of the real depth. This 
is the reason why a spoon appears to be bent when half immersed in water. Every 
point immersed in the water appears to be only f as low down as we expect it to be. 



150 



Science for the Citizen 



the two opposite faces are also parallel. If the faces are inclined, like those of 
a prism, the emergent ray is bent inwards. For reasons shown in Figs. 80 and 81 
the value of R is related in a simple way to the inclination A of the two faces 
through which the beam of light passes, and to the angle D, which the emergent 
ray makes with the ray entering the prism when both are inclined at the same 
angle (x) to the faces from which they respectively emerge or at which they 
enter the prism. This relation is : 

_ sin KA -I- P) 
sin JA 




FIG. 80 

Refraction of pure light of one colour through a prism of 60. The refractive index 
(R) of the glass is 1 5. So if the incident ray (in air) is 45% 

sin r = sin 45 -:- 1 5 = 707 -f- 1 6 = 47. 
Since sin 28 = 0-47, the angle of refraction at the first face is 28. 

To find the angle D for a prism with faces inclined at an angle A an instru- 
ment called a spectrometer is used. This consists of a source of light, which 
projects a thin beam on to the prism and telescope, both capable of revolving 
on a turntable with a graduated scale to record the angle. All that is necessary 
is to find the position in which the telescope and the source of light point 
at equal angles to the two prism faces while the telescope receives the beam. 
To find the refractive index of a solid substance, a prism made of it is used. 



Spectacles and Satellites 

To find that of a liquid a hollow prism with thin glass sides is filled with it. 
The precision of the law can be tested by the consistency of results obtained 
when prisms with faces inclined at different angles are used. 



180 




FIG. 81. ACCURATE DETERMINATION OF REFRACTIVE INDEX. DIAGRAM OF 

SPECTROMETER 

The prism (upper figure) is set so that the two faces which enclose A are equally 
inclined to the line joining to 180 on the scale. The source of light and the telescope 
are moved till the telescope receives the beam while they are inclined at equal 
angles a to this line. The angle of deviation (D) is then 180 2a 

The beams entering and leaving the prism, then make equal angles (x) with the two 
faces which enclose the angle A a and i = 90 x 

Since the / D the sum of the two interior opposite angles^ each (i r) 
D = 2(i - r) /. i = 4D + r. 

Since the /_ A and the two base angles^ each equal to x -f (i r), make up 2 right 
angles 

.'. A -f- 2(* + i - r) * 2(90) 

/. A -f 2(90 -, + ,_,) = 2(90) 



sin_g 
sinr 



sinj(A + D) 
sin A 



DESIGNING MIRRORS 

The social conduct for which the laws of reflection and refraction provide 
us with guidance includes designing mirrors for shaving, or as reflectors for 
lamps, for searchlights, for telescopes and for periscopes; prescribing lenses 
for spectacles ; or combining lenses in cameras, in magic lanterns, in telescopes, 
in microscopes, and in various surgical devices like the laryngoscope and 



152 



Science for the Citizen 



auroscope; also designing prisms for spectroscopes, for periscopes and for 
binoculars. Designing lenses of the right curvature to produce magnification 
involves a knowledge of refraction at curved surfaces, in contradistinction to 
refraction at a plane surface described in its simplest form by SnelTs law. 
The nature of refraction at curved surfaces is more easily understood 
when we know how magnification can be produced by reflection at curved 
surfaces. 

The image produced by a plane mirror cannot be caught on a screen. It 
is a virtual image. To represent it in a scale diagram we have only to remember 




FIG. 82 

I is called the virtual image of the object O, because it merely represents the point 
from which the reflected rays entering the eyeball appear to diverge. This point lies 
as far behind the mirror as the object does in front of the mirror. A virtual image 
cannot be caught on a screen. A real image (pp. 156, 157) can. 

that the only "rays" of light visibly reflected from any point on an object 
(Fig. 68) are those which do not diverge by more than the width of the 
pupil where they enter the eye. This means that to represent the image of 
any point of an object placed before a mirror we only need to follow the 
path of any two rays which enter the eye and trace them backwards to the 
point from which they appear to diverge. Using this method (Fig. 82) shows 
us that the image of a point at a certain distance in front of a mirror is situated 
at the same distance behind it. By tracing out the course of two rays from 
each of the three corners of an L-shaped figure, as in Fig. 83, we also see 
that the virtual image is upright and reversed from left to right. To put the 



Spectacles and Satellites 



153 



issue in another way, the image of a page of print seen in a mirror resembles 
the appearance we should see if we were looking at the script through a 
transparent page. The method of reconstructing the image by representing 
small beams of light or "rays" reflected from its edges as geometrical lines 
therefore agrees with the three most familiar characteristics of reflection 
from a flat surface like a hand mirror or a pool of water. 

In applying the same device to mirrors with curved surfaces such as form 
a slice from the surface of a sphere, the Alexandrian astronomers used the 
reasoning which they had learned to use in dealing with the observer's 
horizon. We can look on the immediate neighbourhood of the place where a 





FIG. 83 

By considering the points from which any pair of rays appear to diverge when an 
angular object is placed before a mirror we can see why the image is reversed from 
left to right. (The rays as drawn diverge more than would be possible for any pair of 
rays entering the pupil unless the object were very small compared with the eye 
placed very close to the object. The exaggeration is necessary to make the relations 
between object and image clear.) 

ray of light strikes a spherical surface as a little patch of flat earth. That is 
to say, the ray is reflected as it would be if it struck a flat mirror placed in the 
plane which is tangential to the surface at the point where it strikes. A line 
which joins the point where the tangent plane grazes the surface to the centre 
of the imaginary sphere from which a concave or convex mirror is supposed 
to have been sliced off as in Fig. 84 must strike the tangent plane at right 
angles. If a ray does so, the angles of incidence and reflection are equal. So 
a ray passing along the line which joins any point on an object to the "centre 
of curvature" of a spherical mirror is reflected back on its own path. Any 
other one is reflected in the same plane so that the angle the reflected ray 
makes with the line joining the point, where it strikes to the centre of curvature, 
is equal to the angle which the incident ray makes with the same line. Rays 
that fall parallel to the line which joins the centre of the mirror to the centre 
of curvature are reflected so that they either converge towards (concave 



154 



Science for the Citizen 





FIG. 84 

A concave or convex mirror can be looked on as a slice from a sphere silvered or 
polished on the inside or outside respectively. The line joining a point on the surface 
of a sphere to its centre is perpendicular to the tangent. So the incident and reflected 
ray make equal zenith angles with the line which joins the centre of curvature to the 
point from which the incident ray is reflected. 







Ccnica,\ 




Convex 



FIG. 85 



The two rules which suffice for representing the image formed by an object placed 
before a concave and convex mirror are : 

(1) A ray (a) which strikes the mirror so that it passes through or is directed towards 
the centre of curvature is reflected back along its own path. 

(2) A ray (b) which runs parallel to the optical axis of the mirror is reflected back 
so that it passes through (concave mirror) or appears to come from (convex mirror) 
the focus. Conversely, a ray (d) coming from the focus (concave mirror) or proceeding 
towards it (convex mirror) is reflected parallel to the optical axis. The optical axis is the 
line which joins the centre of curvature to the centre of the mirror. 



Spectacles and Satellites 155 

surface) or appear to diverge from (convex surface) the neighbourhood of a 
point called the Focus between the centre of curvature and the mirror itself. 
If the distance between the centre of curvature and the mirror is large com- 
pared with the diameter of the latter, the distance of the focus from the 
mirroy (the focal length) is approximately half the distance (radius of 
curvature) from the mirror to the centre of curvature as in Fig. 85. 

If the curvature of a concave or convex mirror is measured we can there- 
fore reconstruct the position of the image when the object is placed at a 
measured distance from it by tracing the path of two rays from each of 
several points at the boundary of the object. For practical purposes the two 
end points are enough. One ray from each point is in the same straight line 
which joins it to the centre of curvature. This is reflected back along its 
own path. The other is parallel to the optical axis and is reflected back so 





FIG. 86 

Rays parallel to the optical axis converge to the focus of a concave mirror which can 
therefore be used like a lens as a burning-glass. Rays which come from the focus are 
reflected parallel to the optical axis. If they come from a point between the focus 
and the mirror they diverge on reflection. This is how mirrors are used on lamp 
reflectors or in headlights to produce a diverging beam of light. 

that it diverges from (convex mirror) or converges towards (concave mirror) 
the focus (Fig. 86). 

The image produced by a convex mirror has the same general charac- 
teristics wherever the object is put. It is an upright virtual image reversed 
from left to right like the image formed at a plane surface, differing only in 
so far as it is always smaller (Fig. 87, a) than the object. This compresses 
a larger visible field into a smaller space. So such mirrors are used as reflectors 
for motor-cycles and cars to display objects approaching from behind. If an 
object is placed between the focus and surface of a concave mirror (Fig. 87, t) 
an upright virtual image reversed from right to left is also obtained. Instead 
of being diminished the image appears to be larger than the object. Everyone 
who has used a shaving mirror, which belongs to this class, knows that the 
image enlarges as the face recedes till a certain distance is reached when 
no clear image is seen. This point is the focus. If an object (Fig. 88) like a 
candle burning in a darkened room is placed beyond the focus of a concave 
lens tilted a little, we can focus a real and inverted image on a screen, e.g. a 



156 



Science for the Citizen 



piece of paper. If the object is between the focus and the centre of curvature 
a clear enlarged image is obtained when the paper is placed a certain distance 
beyond the centre of curvature. If the object itself is placed beyond the 
centre of curvature the screen must be held in some position between the 
centre of curvature and the focus to get a clear image, which is then seen to 
be smaller than the object. Concave mirrors are used in astronomical tele- 
scopes. The object is at a great distance. So an image is formed very near 
the focus, where it is studied with a lens combination called an eyepiece. 



C- -"' P, 



(a) 




FIG. 87 

(a) The virtual upright diminished image produced by putting an object before a 
convex mirror as in using one for sighting objects in the rear of the motor-cycle. 

(b) The virtual upright enlarged image when an object is placed within the focal 
distance of a concave (e.g. shaving) mirror. 

By applying the two simple rules given in the legend of Fig. 85 we can 
make a scale diagram (Fig. 89) of the position of the real image and the 
object, and thus calculate the centre of curvature of the mirror or its focal 
distance. Having done this we can make a scale diagram for the size and 
position of the image when the object is placed in another position. The 
results can readily be checked. Having justified the method we are not bound 
down to drawing a scale diagram to get similar results. The geometry of the 
scale diagram shows us how to calculate the magnification and position of 
the image by the simple formulae explained in Fig. 90. From these formulae 
we can deduce what curvature is necessary in designing a mirror to give a 
suitable magnification at a convenient distance for shaving. 



Spectacles and Satellites 



157 






FIG. 88 

The real inverted image of an object placed beyond the focus of a concave lens may 
be enlarged if the object is within the radius of curvature, or diminished if beyond. 




FIG. 89 

The real image produced when an object is placed between the focus and centre 
of curvature of a concave mirror. If the arrows are reversed the figure also illustrates 
the case where the object is placed beyond the centre of curvature. Most optical 
diagrams are reversible in one way or another. 



158 



Science Jar the Citizen 




B 



FIG. 90. FORMULA FOR MAGNIFICATION AND POSITION OF IMAGES 

In this diagram C is the centre of curvature and F the focus of a concave mirror. Its 
radius of curvature r is equivalent to OC and its focal length is equivalent to OF. 
When the object is between O and F a negative sign which precedes the numerical 
value of v indicates that the distance v is measured to the left of O. 



Size of object AB = XO (approx.) 

Size of image == DE 

Distance of object OB = u = CB -f 2/ 
Distance of image OD = v = 2/ -~ CD 



DE 
Magnification = -^5- 



(i) 



AB 
CB 



tan a = 



DE 



DE 
AB 



DC 
DC 
CB : 



DE , XO , AB 

DF = tan b ^ OF (apprOX - } = OF 



Combining (i) and (ii) 



u-2f 
v-f 2f-v 



DE 
AB 



DF 
OF : 



f ~ u-2f 
uv fv = uf 



uv ~uf - 2fv 



= 2/ 2 -fi, 



(Hi) 



Formula (iii) which gives the position of the image if that of the object and the focal 
length (ir) are already known is easier to recall, if written: 

11 1 . . 



The linear magnification (ii) is (v f) ~ f = j I ; and since (iv) may also be 



v u + 
written as : j = 

f u 



v , . u 

the magnification is : 



u u 

If the mirror is convex OF is measured to the left of O, and a similar figure shows 
that formula (iii) holds when a negative sign is attached to the right numerical value 
for/. 



COMBINING LENSES 



The rules which have been given for finding what sort and size of image 
is produced by an object placed at a known distance from a mirror, or where 
the image is situated, can all be demonstrated by means of a simple optical 



Spectacles and Satellites 



159 



bench (Fig. 91) which can also be used to make similar experiments with lenses. 
Most lenses used in optical instruments are of two general classes (Fig. 92), 
diverging lenses, which direct the path of a beam parallel to the optical axis 



or lamp 
(Object) 



U%kt 



Holder 
far lens 
or mirror 





FIG. 91 

Home-made bench for measuring position and size of real image formed by 
converging lens or (with holder and lamp reversed) concave mirror. 






FIG. 92 

Diverging lenses: (a) planoconcave; (6) biconcave." 
Converging lenses : (c) planoconvex; (d) biconvex. 

F is the focus in each case. 

away from a point called the focus situated on the same side of the lens as 
the source of light, and converging lenses, which bring the rays to a focus 
on the side remote from the source, as when we use a lens to burn a hole 



i6o 



Science for the Citizen 



in a piece of paper. One aspect of a converging lens is always convex. The 
other aspect may be convex, flat, or concave. If concave, the curvature of 
the concave side is less than the curvature of the convex. One side of a 






FIG. 03 

A simple Search-light. The lamp is at the focus of the concave mirror which hence 
reflects a parallel beam on the inner face of a diverging lens whose focus is at F. 







FIG. 94 

Above, the short-sighted (myopic) eye rectified by means of a diverging lens. Below, 
the long-sighted eye (hypermetropic) rectified by means of a converging lens. A 
third common defect "astigmatism" is due to unequal curvature of the refractive 
surfaces in different planes. This is corrected by a cylindrical lens, i.e. a lens cut from 
the side of a cylinder, so that it produces convergence in one plane, but not in the 
plane at right angles to it. 

diverging lens is always concave. The other may be flat or convex with a 
curvature less than that of the concave side. 
Converging lenses, as their name suggests, make the rays of a beam con- 



Spectacles and Satellites 



161 



verge as they emerge. Diverging lenses do the opposite. The eye, as Al 
Hazen was first to recognize, is a tiny camera with a translucent lens which 
produces an image on the sensitive layer or retina. When it functions pro- 
perly it can be focussed for near and far objects by muscles which change 
the curvature of the lens. This adjustment is rarely perfect, and most people 
are a little "short-sighted" or "long-sighted." In short-sighted people light 
from a distant source is brought to a focus in front of the retina, and a 




FIG. 95 

This shows that a ray striking the edge of a convex surface is bent towards the optical 
axis (converging lens) after refraction. To make the construction all we need to know 
is the "refractive index" (R) of the glass. This is taken to be 1 5. Thus if the incident 
angle in air is 45, 

sinr = sin 45 ~ 1-6 = 0-7071 -r 1-5 

Hence r is found to be 28 from tables of sines. Notice that the direction of the ray 
which strikes the centre of the surface of the lens is unchanged when it emerges. 
If the lens is thin, with a large radius of curvature, this ray which strikes the centre 
of the surface may be considered to pass approximately through the centre of the 
lens itself. 

diverging lens is necessary to make the rays converge more gradually as in 
Fig. 94. In long-sighted people, whose difficulty does not arise from the 
common failure of accommodation at an advanced age, the reverse is true. 
The eye is too short, and a suitable converging lens makes rays which enter 
it converge on the retina. 

Why light is bent in this way, when it is refracted at a curved surface, 
is seen in the next two figures (Figs. 95 and 96). A geometrical device similar 
to that for curved mirrors can be used to trace out the path of a ray through 
a lens. The angles of incidence and refraction are measured from the line 

F 



1 62 



Science for the Citizen 



drawn perpendicular to the tangent at the point where the ray strikes the 
curved surface. The use of a lens as a burning-glass shows that parallel 
rays are brought to a sharp focus by a good lens of small curvature, and 
Figs. 95 and 96 show that the direction of any ray which strikes the centre 
of a lens is not changed when it emerges. If we know the focal distance of a 
lens the position and size of an image can be reconstructed by tracing the path 
of two rays from any point on the object. One is a ray which is parallel 
to the optical axis of the lens before it strikes the latter. This ray passes 




FIG. 96 



This shows why a ray striking the edge of a concave surface is bent away from the 
optical axis (diverging lens) after refraction. The construction is made in the same 
way as that of the previous figure. 

throitgh the focus of the lens if the latter belongs to the converging type. 
Otherwise it appears to diverge from the focus. The other ray passes through 
the centre of the lens. Its direction remains unchanged. 

Diverging lenses are like convex mirrors. The image (Fig. 97) is always 
upright, and diminished in size. It is a virtual image. That is to say, it appears 
to be on the same side of the lens as the object and cannot be focussed on a 
screen. An upright virtual image is also obtained with a converging lens if the 
distance between the object and the lens is less than the focal distance of the 
latter. Instead of being diminished it is then magnified. This is what happens 
when we look at small print with a "magnifying glass." If an object is placed 
on one side of a converging lens, we can also focus a real image of it on a screen 
placed on the other side of the lens when the distance of the object from the 
lens is greater than the focal distance. As with a concave mirror, the real image 



Spectacles and Satellites 



163 



is inverted, enlarged if the distance of the object is less than twice the focal 
distance and diminished if the distance of the object is more than twice 
the focal distance (Fig. 98). A camera is simply a box with a converging lens 
placed at the correct distance to focus a real image of distant objects on a 




*:.. 



V 



FIG. 97 



Above, the image formed by a diverging lens. This is always virtual, erect, and 
diminished. Below, the image which is formed by a converging lens when the object 
is not further away than the focal distance of the lens. The image is then virtual 
erect, and enlarged. 




FIG. 98 

When the object is separated from a converging lens by a distance greater than the 
focal distance, an image is formed on the opposite side of the lens and can be caught 
on a screen suitably placed. This real image is always inverted. If the object lies beyond 
2F the image is diminished. If the object lies between F and 2F the image is enlarged. 
This is the principle on which the "magic-lantern" or cinema works. 



photographic plate. For landscapes, the distance of the lens need not be 
adaptable. All rays from a very distant source are practically parallel, and 
converge so that the image is formed very slightly beyond the focus of the 
lens. So the distance between lens and plate is practically equivalent to the 



1 64 



Science for the Citizen 



focal distance of the lens. To find the focal distance of a convex lens, we only 
have to find where a screen must be put on one side of it in order to get the 
clearest image of a very distant object. 

Having found the focal length of a lens we can calculate the position and 
size of the image from that of the object, or conversely we can find the focal 
length of a lens by measuring the position and size of an object and its real 




FIG. 99. FORMULAE FOR MAGNIFICATION AND POSITION OF LENS IMAGE 

The positive sign signifies that a distance is measured from the lens on the same side 
as the object. Hence, if/ is the focal length of the lens, PF =- / 



Size of object AB =- o - CP 
Size of image DE = i 



Distance of object = BP = -f u 
Distance of image DP _ v 



/N 

(i) 



CP 





(ii) =- tan a = 



DE 



v-f 



Combining (i) and (ii) 



f 



V f V 

:. uf ~ uv + vf 



vf uv uf 



. 
This formula for position of the image is easier to recall, if written : 

1 _ l - A 
v u f 

._/ or, from (ii), ~ - - 



Linear magnification is ~ j 



image, by a simple scale diagram like the ones shown in Figs. 97 and 98. 
Alternatively, the calculation can be made from a formula which can be simply 
deduced from the geometrical relations of the diagram, as shown in Fig. 99. 
The truth of the rules applied in either way can be easily established by 
simple experiments with an optical bench like the one in Fig. 91. Their prac- 
tical use lies in finding what magnification can be got from combinations of 
lenses in instruments like the telescope and microscope. 



Spectacles and Satellites 



165 



Ground- gZa^s screen. 




Holder for / to cabch 


Holier far 


eyepiece | 


first image 


object glass 
\ 



distant 

object 

(Lamp) 



of Optical Bcnc?r fc> slicw \\&w TsJcplcrs teles-cope 





"Y 



FIG. 100. KEPLER'S TELESCOPE 

Rays coming from the distant object converge to points on the real inverted image 
which would be seen if a screen were placed between the objective and eyepiece. 
From each of these points a cone of rays then diverges again. From these we can 
select two for graphical representation, one going through the centre of the eyepiece 
unbent, one parallel to the optical axis ; so do not be misled into thinking that the 
rays drawn from the objective to the first image are bent at the latter. The ray passing 
through the centre of the eyepiece would not exist in a telescope having the precise 
proportions shown. It is put in as a sample to find the apparent origin of the rays that 
do exist, i.e. the virtual image. The focus of the objective is F . That of the eyepiece 
is Fe. In the figure the object is not very distant and the real image is formed be- 
yond Fe. Where the object is at a considerable distance the real image falls almost 
at F and the lenses are brought together so that F and K practically coincide. The 

magnification is the ratio of the angular diameters : = -- . If the angles are small, 
as stated on p. 138, 

tan 



\a tan \a 

When F and F* are very close, so that the first image is only just beyond the focal 
distance of the objective and just inside the focal distance of the eyepiece (lower 
figure) 

tanjfr AB ^ AB = BY 

tan \a ~~ BX * BY ~ BX* 



i.e. the magnification is 



Focal distance of objective 
Focal distance of eyepiece ' 



166 



Science for the Citizen 



The simplest combination of lenses to form a telescope is Kepler's 
(Fig. 100). This is made of two converging lenses. One makes the rays of a 
distant object converge very slightly beyond the focus. The inverted real 
image which can be seen if a semi-transparent screen is placed in a suitable 
position between the two lenses is formed in front of the eyepiece at a 



A 




wldch 
fanned 

if the, oyepvzcz 
were, not placed 
between the. oyo, & 
the, 



Virtual 

hnage 

cutfuatty 

seen 



FIG. 101. GALILEO'S TELESCOPE 



(focus of eyepiece) x 
Tc Z 



V _- 




FIG. 102. THE MICROSCOPE WITH SIMPLE EYEPIECE 



distance slightly less than its focal distance. Where the rays would converge 
to form an image if a screen were present, they diverge again in all directions, 
as from the edges of an illuminated object. So the result is the same as if a 
real object were placed within the focus of the eyepiece, i.e. a magnified 
virtual image is produced. Kepler's arrangement therefore gives an inverted 
image. Hence though useful for astronomical purposes it is not suitable 



Spectacles and Satellites 

for viewing landscapes or the actors on a stage. For the reasons illustrated 
in Fig. 100 the magnification is the ratio of the angular diameter (fc) of the 
virtual image and the angular diameter (a) of the object as seen without 
the telescope. This is the ratio of the focal distances of the far lens and 
the eyepiece. 

The simplest type of combination which gives an upright image is Galileo's 
(Fig. 101). This is the type of arrangement used in opera-glasses. The eye- 
piece is a diverging lens. In ordinary circumstances a diverging lens produces 
a diminished upright image which appears to be situated on the same side 
of the lens as the object. This is because rays diverging from the object are 



KrcaZ 
(virtual & erectr) 



Object* 




FIG. 103. THE INVERTING EYEPIECE 

A telescope made of converging lenses can be made to give an erect image by using 
a third lens. If the first real image formed near F 15 the focus of the objective, is made to 
fall at a distance equivalent to twice its focal length from the second lens, a real image 
which is an inversion of the first image, equivalent in size to it, is formed at the same 
distance from the second lens on the opposite side. This image is, therefore, the 
same way up as the object. If it is formed at a distance just a little less than the focal 
distance of the third lens, a magnified virtual and erect image is seen by the eye. 

made to diverge more so that they seem to come from nearer the lens 
(Fig. 97). In the telescopic arrangement the diverging lens is placed between 
the converging lens and the position where the real image would be formed 
if the eyepiece were not there. These rays, converging to the focus of the 
converging lens, are made to diverge so that they appear to come from a 
virtual image larger than the real image which would otherwise be formed. 
This virtual image is an inversion of the latter, and since the latter is itself 
inverted, the result is an upright image of the object seen. 

The microscope (Fig. 102) is essentially like Kepler's telescope. The 
object is placed at a distance from the "objective" greater than but less 
than twice the focal distance of the latter (see Fig. 98), So a real, enlarged, 
and inverted image is formed in front of the eyepiece at a distance less than 
the focal distance of the eyepiece. The real image produced by the objective 



i68 Science for the Citizen 

is seen as a virtual image which is magnified still further. Any two lenses of 
different focal distances can be used as a microscope or telescope according 
as we use the lens of greater or smaller focal distance as the eyepiece. 

A microscope of two converging lenses like a telescope of the same type 
produces an inverted image. One way of avoiding this is the use of a com- 
pound eyepiece (Fig. 103) consisting of two lenses. If one is placed so that 
the focus of the objective is separated from it by a distance equivalent to 
twice its focal distance, the first or real image is replaced by a second reversed 
real image of the same size. The lens nearest the eyepiece gives a magnified 
virtual image of the second real image. If the first real image is formed in 
front of the second lens at a distance greater than, but less than twice, its focal 
distance, the second real image will be larger than the first, so that it will be 
magnified successively by the two lenses of the eyepiece. 



COLOUR FRINGES 

If you make a simple Kepler's telescope or a microscope by combining 
two cheap lenses of different focal distance, as explained, you will have no 
difficulty in discovering why scientists were forced to bother about the nature 
of colour, when instruments to produce high magnification began to be 
manufactured. Any image formed by a combination of ordinary lenses is 
surrounded by a coloured fringe which blurs the outline. The higher the 
magnification, the more troublesome is the distortion which this coloured 
fringe produces. People had long been familiar with the rainbow effect which 
is seen when light shines through pieces of glass cut in the shape of a prism, 
as in Venetian chandeliers and the like. A phenomenon which had hitherto 
been accepted thankfully as an ornament now became a social nuisance. 
Men who were active in advancing the study of astronomy, among whom 
Newton was foremost in the seventeenth century, were compelled to investi- 
gate colours. For the observations which led Newton to advance the views 
now accepted no technique which was not available to the scientists of the 
Hellenistic age was necessary. What was new was a new social need. To make 
good telescopes it was essential to get rid of the coloured fringe. 

Nature, as Bacon taught, can only be commanded when we have first 
learned to obey her. To get rid of the coloured fringe we have to understand 
in what circumstances it is produced. If a parallel beam of sunlight shining 
through a slit (or of a lamp focussed with a converging lens) strikes the face 
of a glass prism, one of two things may happen. It may be bent so that it 
strikes a second face, making an angle with the vertical greater than the 
critical angle, so that it is totally reflected as in Fig. 76. Otherwise it passes 
through the prism, without reflection at one of its faces, as a beam with 
diverging edges. If this diverging beam falls on a plane surface it produces a 
spectrum, a series of bright colours: violet, blue, green, yellow, and red 
(Fig. 104). If part of this coloured beam is allowed to pass through a second 
prism, the only colours which appear in the second spectrum will be those 
which are allowed to strike the second prism. The spectrum cast by a prism 
can be recomposed into white light by passing it through a second inverted 
prism (Fig. 106) or by bringing the rays to a focus with a lens. 



Spectacles and Satellites 169 

Such simple experiments show that ordinary white light can be decom- 
posed into different sorts of light, which we recognize by their colour. What 
our senses recognize as white light is not merely complex. It is not neces- 
sarily built up in the same way. It can be produced by combining the pure 
coloured lights of the spectrum in various ways. Superimposing the pure 
coloured lights of the spectrum (with the right degree of brightness) leads 
to surprising results. The combination of blue and yellow gives white. So 
what we recognize as white light may be made of all the coloured lights of 
the spectrum, or of a mixture of blue and yellow alone. Green and red give 
yellow. So what we recognize as yellow may be either pure spectral light 





Second 
~Prisin 

Tirsf 
Prism, 

Tirst Screen, 

I Slit to admit withstit&r 

sunliglit rtticus yM r & s 

r* 7 G 
or lens- 

FIG. 104 

Newton's classical experiment in which a spectrum was formed on a screen with a 
slit, which could be adjusted to admit only light of a particular region (e.g. yellow) 
of the spectrum. The colour of the pure "monochromatic light" which passed through 
the slit could not be changed by a second prism. 

which cannot be decomposed, or a mixture of green and red, which can be split 
into its constituents with a prism. Blue and red give magenta. Magenta and 
green give white. 

These combinations are not what we should expect from the visible results 
of mixing paints. For instance, blue and yellow dyes usually produce green 
when mixed. If we let light pass through a solution of blue dye before it 
strikes a prism it is nearly always found that the spectrum consists of a 
certain proportion of green rays as well as of blue. The dye fails to transmit, 
i.e. it "absorbs," red and yellow rays. A convenient arrangement for examining 
light transmitted through a prism is called a spectroscope. A yellow dye need 
not be yellow because it absorbs all light except spectral yellow. It may be 
yellow because it transmits red and green, absorbing all other colours. When 
such a yellow is mixed with blue the blue half of the mixture absorbs yellow 
and red rays. The yellow half absorbs blue rays, leaving nothing but green 
to be transmitted. In the same way a visibly red substance may be red, like 

F* 



170 Science for the Citizen 

the ruby lamp of the dark-room, because it absorbs all rays except those at 
the red end of the spectrum, or it may also be red because, like blood, it 
absorbs green light. Of the four principal colours we have left blue, yellow, 
and red when green is eliminated. Since blue and yellow give white, the 
effect is red. 

The spectroscope is used in modern chemistry to detect differences which are 
not recognizable by the eye, and so to distinguish substances which at first sight 
seem to be physically alike in spite of their different chemical properties. 
If a man poisoned with strychnine or cyanide were afterwards placed with 
his head in a gas oven, the examination of a single drop of his blood would 
settle whether he had committed suicide. The spectrum of blood, i.e. light 
first passed through blood and then through a prism, has a black patch in the 
green region. Carbon monoxide, the poisonous constituent of coal gas, com- 




FIG. 105 

An arrangement for superimposing two spectra to show the effect of combining 
different kinds of pure coloured light. 

bines with the red pigment of blood, driving out oxygen. The resulting 
combination is also red, but the size and position of the black patch in the 
green region of its spectrum are not the same as for healthy blood. 

You will see from Fig. 104 that when white light passes through a prism 
the rays at the red end of the spectrum are bent less than the rays at the 
blue end. Consequently magnification with a simple lens must always 
produce an image with a coloured fringe, where rays of complementary 
colours do not overlap and neutralize each other. When Newton set out to 
investigate the formation of coloured fringes, glass of quality sufficiently 
good for making lenses was a rarity. Italian glass-makers came to England 
in the middle of the seventeenth century, and flint glass, the best glass 
suitable for making the simple optical instruments used in Newton's time, 
was evolved in England about the same time. Since a lens of one and the 
same material must necessarily produce coloured fringes, the first effect of 
Newton's discoveries about the spectrum was to discourage further attempts 
to improve the magnifying power of the telescope by combinations of lenses. 
Newton designed a lens-mirror combination with which he could see the 
moons of Jupiter and the horns of Venus, and for a generation astronomers 
relied on further improvement by grinding good concave mirrors. During 
the eighteenth century the variety and quality of glass improved. The problem 



Spectacles and Satellites 



171 



which defeated Newton was solved by practical instrument manufacturers, of 
whom Dollond was the successful competitor for the patent rights issued 
in the latter half of the eighteenth century. 

This was possible, because another good quality glass was now available. 
The length of the spectrum produced by prisms of different kinds of glass 
varies considerably. For instance, if made of "crown glass" (calcium with 
potassium or sodium silicates), a prism of some particular size and shape pro- 
duces a much shorter spectrum than one of the same dimensions made from 
"flint glass" (lead and potassium silicates). On the other hand, the magnifying 
power of lenses made of glass of two different kinds does not vary very much. 
This fact makes it possible to get high magnification without "chromatic 





Correcting chromatic aberration of lenses. A second prism placed as in figure 
neutralizes the dispersion of white light produced by the first. If the first prism is of 
crown glass, a flatter prism of flint glass suffices to neutralize the dispersion. In the 
same way a diverging lens of flint glass can neutralize the coloured fringe (RV) pro- 
duced by a converging lens of crown glass. A flint glass lens which does this will be 
one of low diverging power as compared with the converging power of the crown 
glass lens. Hence the combination is itself a converging lens. 

aberration/' i.e. formation of coloured fringes which blur the outline of the 
image seen through a telescope or microscope. Modern instruments use 
Dollond's "achromatic" lenses formed by sticking together a crown glass 
converging lens of high curvature and a flint glass diverging lens of low 
curvature. The curvature of the flint glass lens is sufficient to neutralize the 
spectral "dispersion" of the crown glass lens in the same way as one prism 
can neutralize another (Fig. 106), without being sufficient to neutralize the 
magnifying power of the combination. 



INTENSITY OF LIGHT 



So soon as people were forced to an active interest in the nature of colour, 
the problem of measuring the intensity of a source of light acquired a new 
importance. If we match things by daylight our judgments do not agree 
with the result of matching the same things by artificial light. Nowadays 



IJ2 Science for the Citizen 

we explain this by saying that electric light or gas light contains more red 
or yellow light than sunlight. The conditions of urban life in northern 
climates, the multiplication of sources of illumination and artificial dyes 
have compelled us to set up a standard of measurement by which we can 
tell how much light we get for the money we spend on illuminating our 
streets and dwellings, and how different shades of pigment harmonize. 
The principle used in determining the candle power of an electric light bulb 
is a simple application of Alexandrian optics. The only evident reason why 
antiquity did not bother to measure light is that antiquity had no need to do so. 
Intensity of ordinary white light is measured nowadays by comparison 
with a standard source of illumination. The first standard set up was a candle 
of particular dimensions and composition. The British standard was a sperm 
wax candle weighing six to the pound and burning 120 grams per hour. 
There are far more reliable sources of light today, and though we use the 




FIG. 107 

A beam of light which diverges from one and the same source illuminates four times 
the area at twice the distance, nine times at three times the distance, and so forth. 
Hence the intensity of illumination or quantity of light falling on unit area is inversely 
proportional to the square of the distance. 

term candle power, the actual physical standard used is not a candle. One 
of the best is a specially constructed burner for a constant mixture of air and 
pure pentane (the chief constituent of gasoline) arranged to give a flame of 
fixed dimensions. This is defined as a candle power often. To find the candle 
power of any other source of light by comparison we compare the distances 
at which the standard and the source of unknown candle power produce 
the same brightness. One way is to place them on opposite sides of a paper 
screen with a grease spot which makes the paper translucent in the middle. 
When the amount of light reflected from the opaque part and transmitted 
through the grease spot is equal on both sides, the outline of the grease becomes 
invisible. Another way is to compare the distances at which two shadows 
cast by the same object on the same screen, when the two sources of light 
are not quite in the same straight line with the object, look equally dark. 
From the elementary principle that "light travels in straight lines," it follows 
(Fig. 107) that the brightness of a source of light, i.e. the amount of light 
which falls on the same amount of surface, is inversely proportional to the 
square of the distance. So, if the distance of the source from the grease spot 
or shadow-screen of the "photometer" is three times the distance of the 
other, and the grease spot is invisible or the shadows are matched, its candle 
power is nine times as great. 



Spectacles and Satellites 173 

INVISIBLE LIGHT 

One of the earliest optical phenomena which attracted interest was the 
intense heat produced at the focus of a concave mirror when a parallel beam 
of sunlight strikes it. One legend credits Archimedes with applying the 
principle in naval warfare by attempting to set fire to enemy ships with large 
metallic mirrors. The analogous experiment for setting fire to a dry leaf or 
a piece of paper with a magnifying glass is one which most of us have carried 
out in our schooldays. The most celebrated burning-mirrors were made about 
the time of Newton's work on the spectrum by Tchirnhausen, who, with a 
large copper concave mirror, used the sun's rays to boil a kettle of water and 
to melt a hole in a coin. 

Such phenomena force us to examine more closely what we mean by light 
rays. Ordinarily we recognize a beam of light by its effect on our eyes. The 
spectrum teaches us that unaided vision does not distinguish between colours 
which are found to be different when examined with a prism. Our eyes are 
not perfect instruments for recognizing when things are alike and when they 
are different. So we have to look for a new way of defining colour. The 
physicist says that things are of the same colour when they absorb rays of 
the same part of the spectrum. If our eyes are faulty as a means of recognizing 
colour, our judgments may also be faulty when we say that a beam of light 
can boil a kettle or blur a photographic plate. Some people are colour- 
blind in the sense that they cannot distinguish between green and red. It 
is easy to show that the distinction which the rest of us recognize is a real 
one. White light shining through a piece of red glass which absorbs all 
rays except red, as does the ruby lamp of the dark room, will not spoil a 
photographic plate. White light shining through a piece of green glass does 
so. Just as the effects which we observe in mixing colours cannot be explained 
by sticking to the belief that two colours are the same if the eye detects no 
difference, so the physical effects which we ascribe to "light" do not corre- 
spond perfectly with what we are able to recognize at first sight. In a certain 
sense we are all colour-blind. 

The discovery that this is so was not made till more than a hundred years 
after Newton's work on the spectrum, when the blackening of silver chloride 
in sunlight was first studied. This effect of a visible source of light on silver 
salts is the basis of modern photography. We thus know of two physical 
effects other than what we see directly when light shines. A source of light 
can be used to produce heat or to produce chemical change. Neither of 
these effects corresponds exactly with what we usually recognize as light, 
i.e. the visible limits of the spectrum at the red and violet ends. If we move 
a sensitive thermometer along a spectrum thrown on a screen it detects heat 
above the visible limit of the red end of the spectrum and registers very little 
effect in the visible violet. An ordinary photographic plate is affected very 
little by the red rays, and is blurred beyond the limits of the visible violet 
end of the spectrum. If the spectrum is thrown on the screen vertically, the 
thermometer registers no heat when moved sideways beyond the visible 
limits, which are sharply defined. This points to the conclusion that the 
photographic plate can be affected by light which is more highly refracted 



174 



Science for the Citizen 



than light which we see directly, and that heating can be produced by light 
which is less highly refracted than light which we see directly. 

To avoid using the word light in an unfamiliar sense, you may prefer to 
speak of three kinds of "radiation," visible radiation, invisible "infra-red" 
radiation which is recognizable by its heating effect, and "actinic" or ultra- 
violet radiation which is recognizable by its chemical effect on silver salts. 
All three sorts of "radiation" have four characteristics of visible light. First, 
they can be communicated through empty space. That is to say, the effects 
which we describe as characteristic of a certain kind of radiation occur when 
the source is separated by a vacuum from the thing it influences. Second, 




If a taxik of wotcr, 
or slab of thick 
glass , is pl&cecL her? 
practLCalL/- no result 
is obtain&d. 



FIG. 108 

A hot metal ball is placed at the focus of a concave metallic mirror. The black bulb 
of a sensitive thermometer placed at the focus of a second mirror registers a rise of 
temperature. In the type of thermometer shown (an "air thermometer") the two 
blackened bulbs contain air. The expansion of the air in one bulb forces the fluid in 
the corresponding limb downwards. Unless the metal ball which is the source of 
radiation is nearly incandescent, no effect will be registered while a slab of glass is 
held between the two mirrors. 

they display the phenomenon of refraction, as the spectrum experiment 
shows. Third, they are reflected according to the same laws as light (see 
Fig. 108). Finally, they are obstructed by a black surface. If actinic rays fall 
on a black surface they are not reflected or scattered. Hence the image of a 
black object does not darken silver salts. That is why the photographic plate 
is a "negative." The absorption of heat rays is illustrated by the arrangement 
on the left-hand of Fig. 109, which also shows (right-hand) that a black sur- 
face emits heat radiation better than a white one. 

The absorption or obstruction of radiation by a black body is always 
accompanied by rise of temperature, the effect being specially pronounced 
in the red end of the visible spectrum and the region of the radiations which 
are less highly refracted. We call the infra-red rays heat rays when the pro- 
duction of heat is their most striking effect. Nowadays, we know chemical 



Spectacles and Satellites 



175 



reactions which are sensitive to these rays, just as the silver bromide of the 
ordinary photographic plate is sensitive to actinic rays and to visible rays in 
the blue end of the spectrum. Such reactions form the basis of infra-red 
photography. Aside from the specific physical effects mentioned, and the 
extent in which they are refracted, different sorts of radiation differ con- 
siderably in the ease with which they pass through different substances. 
Some kinds of glass are comparatively opaque to the invisible ultra-violet 
rays, which affect the photographic plate. This fact is of biological importance 
because some chemical reactions which occur in the animal body depend on 
ultra-violet light. The invisible heat rays of the infra-red spectrum pass 
through glass, otherwise we should not be able to recognize them. There 



Q 





FIG. 109 

A metal box filled with boiling water can be used as a source of heat radiation. If the 
box is uniformly white, the black bulb of an air thermometer of which the other 
bulb is unblackened registers greater absorption of heat. If one face of the box is 
black more heat is radiated from it. 



are other heat rays which do not pass through glass. A body which is heated 
to incandescence soon ceases to give off heat rays which pass through glass 
when it cools beyond the temperature at which it just ceases to be visible, 
although (as the experiment of Fig. 109 shows) heat rays can be detected 
from a metal box filled with boiling water. Thus a greenhouse is a heat trap. 
It admits all radiations which pass through glass. These are absorbed, 
producing a rise in temperature, which leads to the production of heat rays, 
which cannot pass through glass. Dry air is highly transparent to the heat 
rays. Water which permits visible light to pass through it with hardly any 
loss is relatively opaque to invisible heat rays, especially if a little alum is 
dissolved in it. Some substances which obstruct visible light are readily 
penetrated by invisible radiation. The X-rays of medical diagnosis are ultra- 
violet rays which can penetrate the tissues of the human body sufficiently 
well to act upon an ordinary photographic plate. 



ij6 Science for the Citizen 

EXAMPLES ON CHAPTER III 

(These can be answered by making scale diagrams on graph paper) 

1. A concave mirror is 10 inches wide and } inch deep in the centre (neglecting 
the thickness of the glass). How far away must the chin be held to obtain the 
best magnification for shaving? 

2. An extensible camera gives a clear image of a distant landscape when the 
lens is 8 inches from the ground-glass screen. How far must the screen be 
extended to get a good photograph of the page of a rare book placed 2 feet 
from the lens? 

3. At what distance from the lens should the book be placed to get a repro- 
duction of exactly the same size, and at what extension of the plate from the 
lens? 

4. To make a lantern slide of the same object quarter size at what distance 
from the lens must the book be placed, and how far must the plate be from 
the lens ? 

5. A camera extends so that when the lens is 1| feet from the screen it gives 
a life-size image of a bird's egg in the nest. At what length must it be focussed 
to snap a hawk hovering high overhead? 

6. A headlamp consists of an electric light bulb placed at the focus of a con- 
cave mirror, and a diverging lens 9 inches in diameter and of focal distance 
12 inches placed 4 inches in front. By how much will the angular divergence 
of the beam be diminished if the lens is shifted forwards by 2 inches ? 

7. Make a diagram to show the images of a point placed midway between 
two mirrors, and confirm your conclusion by standing between two. How is it 
that the moon's image is replaced by a band of light when the sea is covered 
by ripples? Find the distance between the third and fourth image seen in each 
of two mirrors 10 feet apart with an intervening object 7 feet from one of them. 

8. Draw a diagram of the images formed by an object equidistant between 
two mirrors inclined (a) at 90, (b) at 60. How many images are formed in 
each case? 

9. A glass vessel 8 inches deep is filled with 

(a) methyl alcohol, whose index of refraction is 1-332. 

(b) carbon bisulphide, whose index of refraction is 1-63. 

(c) Canada balsam, whose index of refraction is 1-52. 

(d) ethyl ether, whose index of refraction is 1 352. 

What is the apparent depth in each case? 

10. What is the apparent maximum north polar distance of any star ever 
visible to a fish in the River Thames (Lat. 51)? 

11. If the index of refraction from empty space to air is 1*0003, find the 
true elevation of the sun above the horizon plane when the observed altitude 
is 45. (Assume that the earth is approximately flat.) 



Spectacles and Satellites 177 

12. If a long-sighted person cannot see objects nearer than 50 cm. dis- 
tinctly, find the focal length of a spectacle lens which will enable him to see 
objects as near as 25 cm. (clue the lens must be capable of forming an image 
at 50 cm. of an object placed 25 cm. from it). Will the lens be converging or 
diverging? 

13. A short-sighted person cannot see clearly beyond 30 cm. from the eye. 
What kind of lens must be used to enable him to see distant objects, and 
what will be its focal length? 

14. A myopic patient can see print best at 12 cm. Find the focal length of 
a spectacle lens to extend his range to 30 cm. 

15. In the practice of the optician the power of a lens of 1 metre focal length 
is said to be one diopter, that of a lens of 50 cm. focal length 2 diopters, etc. 
The -f sign indicates a converging, the sign a diverging lens. Give the 
results of the three last examples in diopters with the appropriate sign. 

16. Explain why the sun looks red in a fog. How would the penetration of 
an arc lamp be affected by enclosing it in red glass? 

17. If a prism is made of crown glass whose refractive index is 1-523, to 
what angle must it be ground to give a minimum deviation of 25? 

18. With the spectrometer method of Fig. 81 the minimum deviation for 
sodium light is 29 32' with a prism whose angle is 44 46'. Find the refractive 
index of the glass for sodium light. 

19. White light falls at right angles to one face of a prism whose vertex is 
25. For the end rays of the spectrum the glass of which it is made has refractive 
indices 1-61 and 1-63. What is the angle of divergence between the visible 
limits of the spectrum? 

20. If the light of the sun is passed through a small hole on to a screen, an 
image of the sun is formed, but if the aperture is a large one, an image of 
the aperture is formed. Make a diagram to explain this. In a pinhole camera 
the screen is placed 6 inches from the hole. If the camera is at a distance of 
50 feet from a tree 20 feet high, and is on the same level as a point half-way up 
the tree, what will be the height of the image? 

21. Find the number of candles which at a distance of 420 cm. will give 
the same illumination as one candle of the same make placed at a distance 
of 60 cm. 

22. If two electric glow lamps of 30 and 16 candle power are placed 120 cm. 
apart at the same height, at what points on the line through their centres do they 
give the same illumination? 

23. By means of the grease-spot photometer (p. 172) the intensities of two 
glow-lamps are compared. The outline of the grease spot disappears when 
the photometer is 83 cm. from one lamp and 53 cm. from the other. On inter- 
changing the lamps, and adjusting the photometer till the grease spot disappears, 
the distances are now 50 cm, and 77 cm. Find the ratio of the intensities 
lamps. 



iy8 Science for the Citizen 

24. The illumination produced by the light of the full moon falling per- 
pendicularly on a screen is the same as that of a standard candle at a distance 
of 4 feet. What is the candle power of the moon, its distance from the earth 
being 240,000 miles? 



THINGS TO MEMORIZE 

1. Law of Reflection. Angle of incidence = Angle of reflection. 

2. Snell's Law of Refraction. Sin i =- R sin r 

1112 

3. Spherical Mirrors. + ~ - =- 

v u f r 

Linear magnification - . 

u 

I 1 _ 1 v 

4. Lenses. -7. Magnification = - 

v it f u 



CHAPTER IV 

CHILDREN OF THE SUN 

The Decline of Mere Logic 

THE year 1543 was notable in the history of human knowledge for the 
publication of the De Revolutionibus by Copernicus and the De Fabrica Humani 
Corporis by Vesalius. One marks the beginning of a new epoch in man's 
understanding of inanimate nature, the other marks the beginning of a new 
epoch in man's understanding of his own nature. There were abundant reasons 
why the opening years of the sixteenth century of our own era should have 
been signalized by a great advance in the study of the heavens. In the three- 
quarters of a century which preceded the work of Copernicus, navigation 
had rapidly attained a level far above any of the achievements of antiquity. 
Mechanical clocks were becoming available for astronomical observatories. 
Printing made possible the distribution of new information and old sources. 
In this situation a more exact knowledge of the position of the planets had 
an immediate practical importance which has been explained in Chapter II 
(p. 106). Although mechanical ingenuity had solved the problem of making 
standard (e.g. Greenwich) time portable in countries where sunlight is 
scarce, the clock was as yet and was to remain for a long time to come 
incapable of recording standard time over a long voyage. So measurement 
of longitude was still contingent on more precarious sources of information, 
as, for instance, the occultation of a planet by the moon's disc, symbolically 
represented by the Turkish national emblem. 

The view which put the sun at the centre of the solar system was not 
new. It had been anticipated by Aristarchus perhaps likewise by the 
Pythagorean brotherhoods a century earlier. It had been rejected by Hippar- 
chus because there was no direct evidence for the annual parallax of a fixed 
star or for the earth's diurnal rotation. The parallax of a fixed star was not 
detected till three hundred years after the death of Copernicus, and the 
retardation of the pendulum at low latitudes (see p. 288), the first terrestrial 
experience pointing to the earth's axial rotation, was not recorded till fifty 
years after Kepler's successful exposition of the heliocentric doctrine. In this 
chapter and the next one we shall see why the Copernican view was bound 
to engage a sympathetic hearing among those equipped to understand it, 
in spite of the absence of new evidence to meet the seemingly decisive 
objections which could still be urged against it. 

The invention of wheel-driven portable clocks made the determination 
of longitude at sea a technical possibility which began to be recognized in the 
fifty years that preceded the work of Copernicus and became a topic of absorb- 
ing interest in the half century which followed its publication. Before the 
modern chronometer came into use the two most simple methods were the 
observation of eclipses and occultations of the planets by the moon's disc. 
Aside from occultations, when the decimation as well as the R.A. of the 



i8o Science for the Citizen 

moon and a planet are identical (within about a quarter of a degree), the 
astrological lore of the medieval world attached considerable importance to 
the times when the R.A. of a planet is the same as that of the moon or sun 
(conjunction) and when they differ by 180 (opposition). The times of 
conjunctions and oppositions were therefore recorded in all ephemerides 
and in almanacs, before tables giving the daily variation of the moon's R.A. 
at a given station were available for the method of lunar distances. 

Amerigo Vespucci (Fig. 110) is said to have found his longitude when 
his ship was in latitude 10 N. from the following observations. At 7.30 p.m. 
by local time, i.e. 1\ hours after local noon, the moon was 1 E. of Mars. 
At midnight (local time) it had travelled to 5| E. of Mars. Thus the moon 
had moved through 4| in the same number of hours. So it would have been 
in conjunction with Mars at approximately 6.30 p.m. local time. On the 

^->- 
-/- / ^-T^ ^ Sun 

MOO7V 





12 p.m. NwvniTxrg 5Q j 

time r ' 

6.30p.m. Ships iunc 

' Vespucci found, his 
' -o I 



FIG. 110. How AMERIGO VESPUCCI FOUND His LONGITUDE BY A CONJUNCTION 

OF THE MOON AND MARS 

When the R.A. of the moon is the same as that of a planet they are said to be in con- 
junction. If their declination is also the same 3 the planet will be occulted by the moon's 
disc. If the declination differs by a small angle it is still possible to gauge when the 
R.A. of the two is the same by the naked eye. The exact moment of the conjunction 
can be determined by successive observations of their local co-ordinates (azimuth 
and zenith distance). From these the R.A. can be calculated by the spherical triangle 
formula given on page 195. 

same date his almanac prepared by Regiomontanus recorded a midnight 
conjunction of Mars at Nuremberg. So when the time at Nuremberg was 
12 p.m. it was 6.30 p.m. at the ship's position. Local time was 5| hours 
behind Nuremberg. Hence, he calculated that the ship was 5| x 15 = 82J 
west of Nuremberg. In addition to this example, Marguet (Histoire de la 
Longitude de la Mer, etc.) cites others. Columbus sought a port to observe the 
time of opposition of Jupiter and the moon in the 1493 voyage. In February 
and April of 1520 Andres de San Martin, the "best trained pilot" of 
Magellan's expedition, observed conjunctions of planets "according to 
the instructions of Faleiro who had composed a treatise on longitudes for 
the special use of this expedition." 

The words in italics show that mapping out the track of the planets was 
no longer a merely academic issue. It was a substantial problem of technology 
in the age of the great navigations. So the announcement of the doctrine of 



Children of the Sun 



181 



Copernicus fulfilled an immediate social need. Not less important is the 
fact that the invention of spectacles proved to be the midwife of an instru- 
ment which weakened the inherent plausibility of the opposing view. The 
telescope revealed the planets as bodies with phases like the moon (Fig 111), 
shining with reflected light like ourselves, enjoying night and day as we do, 
and having moons revolving round them like our own. Observations on the 
sun's spots showed that the sun rotates about its own axis. So there is 
nothing outrageous in supposing that we may do the same. 




FIG. ill. THE PLANETS' PHASES, MARS ABOVE, VENUS BELOW 

Hitherto very little has been said about the motion of the planets. The 
account of the heavenly bodies given in the first two chapters was mainly 
concerned with the apparent motion of the sun and the fixed stars. While 
observing the stars, which seem to maintain the same relative positions in 
the uniform rotation of the heavenly sphere, the priestly astronomers of 
Egypt and Sumeria, and, it may be added, those of the calendar civilizations 
of Central America, recognized other bodies which do not have a fixed 
position in the celestial sphere, nor retreat steadily among the fixed stars in 
one direction like the sun and moon. Five of these bodies, Mercury, 
Venus, Mars, Jupiter, and Saturn, were known to the ancients. The extreme 



182 Science for the Citizen 

brilliancy of two of them Jupiter and Venus sufficiently explains the 
attention which their vagaries attracted. Three of them might be seen at 
some periods on the meridian in the course of the night. At such times 
nightly comparison showed that they seemed to be retreating slowly in the 
direction opposite to the sun's annual or the moon's monthly motion. The 
other two, namely Mercury and Venus, are never seen throughout the whole 
of any night. Each may be seen alternately as an evening star setting within 
three hours after sunset or as a morning star rising shortly before daybreak. 
The brightness of Venus makes it conspicuous in the twilight almost as soon 
as the sun sets and long before the brightest fixed stars are visible. 

Owing to the brightness of the planets it is easy to recognize in 
fact, difficult to avoid noticing that their position among the fixed stars 
changes. During one month a planet may be east of a particular star, and may 
rise or set farther south. Next month it may be seen west of the same star, rising 
or setting perhaps farther north. Thus the R.A. and declination of a planet 
can be seen to change without recourse to measurement. The same times 
of rising and setting or of the meridian transit of any fixed star recur after a 
year. So the history of any fixed star in the course of one year is the same 
as its history in the preceding or succeeding year. This is not true of the 
planets. For instance, if you had watched for Venus month by month during 
1934 and 1935 with the naked eye, you could have recorded its history as 
follows. In January 1934 Venus was a brilliant object setting in the early 
evening sky. At the beginning of February it was invisible. By the beginning 
of March it was a morning star rising within an hour before sunrise. In May 
it was still a morning star rising just before daybreak, and might be just visible 
before daybreak in June, July, August, and September. In October and 
November it was not visible. At the end of December it was visible just after 
sunset, as also in January 1935. In February 1935 it was a bright star in the 
evening sky for about two hours after sunset, remaining a conspicuous 
evening star till August, and in September again invisible (see also Fig. 63). 
The history of Mars during the same period was briefly (Fig. 112) as follows : 
in January and February 1934 Mars might be just visible for a short while 
after sunset. During March, April, May, and June, it would be hardly visible 
at any time. In July it would be visible before sunrise in the early hours, 
rising soon after midnight from August to December, but never on the 
meridian before the morning twilight. In January 1935 it rose before mid- 
night and crossed the meridian before morning twilight. By April it was 
rising before sunset, crossing the meridian about midnight. By the end of 
June it was setting just before midnight, and had passed the meridian at 
sunset. It remained an evening star, being still just visible in twilight after 
sunset from September to December. 

Each planet has its own cycle or synodic period in which it gets back to 
the same position relative to the earth and the sun. That of Venus is 584 
days. If you look up Whitaker's Almanack for 1934 and 1935 you will see 
that on March 12, 1934, the R.A. of Venus, then at greatest brilliance, was 
2 hours 37 minutes behind the sun, and it was then a morning star. Its R.A. 
was 2 hours 30 minutes behind the sun on October 13, 1935, 2 hours 38 
minutes on October 18th, and 2 hours 43 minutes on October 23rd. So it had 



Children of the Sun 

returned to its original position with reference to the earth and sun (see 
Fig. 63) on October 17th, 584 days later. The interest which was excited 
by the cycles of the planets in the priestly cultures of antiquity is illustrated 
by the following citation from a recent account of the calendar of the extinct 
Maya civilization of Central America. The Maya calendar contained five 
different long cycles, a year of 365 days, a year of 360 days, a period of 
260 days, a lunar year based on the lunar month, and the Venus cycle. 
According to the source cited:* 



XXTV 



XVUT 




XII 
FIG. 112. THE TRACK OF MARS IN 1934-35 



The planet Venus was the object of an important cult. The revolution of 
Venus occupies a little less than 584 days; five of these Venus years equalled 
eight mean solar years (584 x 5 = 2,920, 365 x 8 = 2,920). The Mayas, 
however, were well aware that the Venus year was actually less than 584 days. 
They knew its length to the second decimal point. The actual period is 583 92 
days, and to correct this error the Mayas dropped four days at the end of every 
sixty-one Venus years, and at the end of every three hundred Venus years eight 
days were dropped. This system was so accurate that had the Maya Venus calen- 
dar continued to function uninterruptedly up to the present day, the error over 
this period of over a thousand years would not have amounted to more than 

* Field Museum of Natural History, leaflet No. 25, pp. 57-8. 



184 



Science for the Citizen 



a day. Such an accurate knowledge of the cycle of Venus, the revolutions 
of which are by no means regular, points to centuries of sustained observations. 
Up to the present, no deity in the Maya pantheon has been satisfactorily identi- 
fied with Venus. In Mexico^ however, Quetzalcoatl was closely associated with 
Venus as the Morning Star. In addition to Venus, the planets Mars, Mercury, 
and Saturn, were closely observed, and their phases accurately calculated. When 
one recollects that the Mayas were dependent solely on the naked eye for 



^jV5un~ 

V" 




\ 
\ 

Venus in \i^ 

tins posi&m~ V| \ 
rises before \^ 



r enus m, 

this position 
sets after 
sunset 

/ 



Abtids 

Vervuus in, position, A is 



sun, 



risen 



*At this IcmMude 
in, position B is atitke 
trot th& 



"?ias nob yet setr. 



Mars 

izi opposition crosses 
meridian zt rnidnwhtr' 

FIG. 113 

The inferior planets, Mercury and Venus, have always passed the meridian when they 
become visible or have not yet reached it when they cease to be visible. Since it is 
customary to represent east on the right-hand side of a map, and also because the 
horizon plane rotates eastwards to meet the sun, the figure is drawn so that the South 
Pole is nearest to the reader. Venus is seen at maximum elongation west of the sun 
as a morning star, and at maximum elongation east of the sun as an evening star. 

their observations, one is astounded at the grasp they had on the movements of 
the heavenly bodies. In various cities regular lines of sight existed for the 
observation of the equinoxes, solstices, and other important points of the 
tropical year, notably at Uaxactun, Copan, and Chichen Itza. 

The precise positions of the inferior planets Mercury and Venus, which 
are only visible as morning or evening stars^ cannot be gauged by the methods 
which we have mentioned in Chapter II. Since they never cross the observer's 
meridian by night (Fig. 113), we cannot find the R.A. or declination of either 



Children of the Sun 

of them by recording the time and zenith distance at meridian transit. To 
trace out the motions of the planets we have to know how to calculate the 
right ascension or declination of a heavenly body from observations upon its 
position when it is not on the meridian. Even for mapping daily the entire 
course of the moon's monthly cycle, the methods which we have used so 
far are not wholly sufficient, because on several days the moon will not be 
visible at its time of transit. It is not necessary to watch for the time of transit 
of a celestial body to determine its co-ordinates in the celestial sphere (R.A. 
and declination). With the help of spherical trigonometry we can find the 
R.A. and declination of a heavenly body, if we know the local time and the 
local co-ordinates (azimuth, altitude, or zenith distance) and latitude. Con- 




FIG. 114. LOCAL CO-ORDINATES OF A STAR 

The horizon bearing or altitude is 90 z.d. The zenith distance z.d. is measured by 
the arc TZ or the flat angle ZOT in the azimuth plane. The meridian bearing or 
azimuth is the arc NQ which in degrees is the flat angle NOQ or the angle between 
the meridian plane NZS and the azimuth plane ZOQ. 

versely, the navigator need not wait for a heavenly body to cross the meridian 
to find his latitude provided that he has a star map, or an almanac giving 
tables of the declination of the stars. 

The local co-ordinates of a star when it is not on the meridian have already 
been defined on page 48, and will be understood with the help of Figs. 114 
and 115. In Chapter II we have seen how to represent the position of a star 
in the heavenly sphere by small circles of declination parallel to the celestial 
equator and great circles of R.A. intersecting at the celestial poles. Such a 
map is true for all places, and relevant to any time of the year. At any fixed 
moment at a particular place we can represent the position of a star by small 
circles of altitude parallel to the circular edges of the horizon and great circles 
of azimuth intersecting at the zenith (Fig. 114). The altitude circles are num- 
bered by their angular elevation above the horizon plane, just as declination 



1 86 



Science for the Citizen 



or latitude circles are numbered by their elevation above the equator plane. 
An azimuth circle is numbered in degrees off the meridian by joining to the 
observer the ends of an arc on the horizon plane intercepted by the meridian 
and the azimuth circle, in just the same way as a circle of longitude is num- 
bered by the angle between the end of an arc of the equator intercepted by 
it, the centre of the earth, and the point where the equator is cut by the 
Greenwich meridian. The azimuth of a star is therefore its east/west bearing 
with reference to the meridian. If you have mounted your home-made 
astrolabe or theodolite of Fig. 115 to revolve vertically on a graduated base 



North, 




FIG. 115. HOME-MADE APPARATUS FOR MEASURING AZIMUTH AS WELL AS Z.D., 

OR ALTITUDE OF A STAR 

The materials are three blackboard protractors (you can make these with a fret- 
work set), a piece of iron tube (gas pipe), and a plumb-line. The object at which the 
instrument points has azimuth 70 East of North. 

set so that points due south or north., the azimuth of a star is the angle 
through which you have to turn the sighting tube (or telescope) on its base, 
and the altitude is obtained by subtracting from 90 the zenith distance. 
If the protractor is numbered reversibly from to 90 and 90 to 0, you 
can, of course, read off the altitude at once. 

In a modern observatory the declination or R.A. of a heavenly body 
can be found when it is not on the meridian with an instrument called an 
equatorial telescope. A simple type of equatorial telescope can be made 
by fixing a shaft pointing straight at the celestial pole and mounting a tele- 
scope (or a piece of steel tube) so that it can rotate at any required angle 
about the shaft itself as axis (Fig. 116). If the telescope is now clamped at 
such an angle as to point to a particular star, we can follow the course of the 



Children of the Sun 

star throughout the night by simply rotating it on its free axis without 
lowering it or raising it. If it is set by very accurate modern clockwork so that 
it can turn through 360 in a sidereal day (i.e. the time between two meridian 
transits of any star whatever), it will always point to the same star. Since it 
rotates about the celestial axis the tilt of the telescope is the polar distance 
of the star, and if the clock is set at (= XXIV) hours when the First Point 




eel. 



FIG. 116 

A simple "equatorial" made with a piece of iron pipe and wood. The pipe which 
serves for telescope rotates around the axis A fixed at an angle L (latitude of the 
place) due north. When it is clamped at an angle PD (the "polar distance" of the 
star or 90 Declin.) you can rotate it about A as the star (S) revolves, keeping S 
always in view. 

of Aries crosses the meridian, the R.A. of the star is the sidereal time at 
which the telescope lies vertically above its axis of rotation. If you compare 
Figs. 116 and 117 you will therefore find little difficulty in seeing how it is 
possible to measure the celestial co-ordinates of a heavenly body at times 
when its transit is invisible, and if you are satisfied that it can be done, the 
next few pages can be deferred till you have read the ensuing sections on 
the hypotheses of Copernicus and Kepler. On the other hand, you will find 



i88 



Science for the Citizen 



it beneficial to work through it, when you have done so, if you wish to 
understand the final section of this chapter on the dating of ancient 
monuments. 



THE SPHERICAL STAR TRIANGLE 

In the time of Copernicus and Kepler it was not possible to make clocks 
sufficiently reliable for the construction of an equatorial instrument, and 



Pole 



Zcrdtk 




Horizon boundary /Western) 




FIG. 117. APPARENT ROTATION OF THE CELESTIAL SPHERE 

The position of a star (T) in the celestial sphere may be represented by a point where 
a small circle of declination which measures its elevation above the celestial equator 
intersects a great circle of Right Ascension. All stars on the same declination circle 
must cross the meridian at the same angular divergence from the zenith and are 
above the observer's horizon for the same length of time in each twenty-four hours. 
The arc PT or flat angle POT measures the angular divergence of the star from the 
pole (polar distance) and hence is 90 Declin. All stars on the same great circle of 
R.A. cross the meridian at the same instant. The angle between two R.A. circles 
measures the difference between their times of transit. The angle h measured from 
the meridian westward between the plane of the meridian and the R.A. circle of the 
star is the angle through which it has rotated since it last crossed the meridian. If 
the angle is 15 it crossed the meridian one hour ago. So // is called the hour angle 
of the star. If the hour angle is h degrees, the star made its transit h ~- 15 hours pre- 
viously. The hour angle is usually expressed in time units. 

the determination of Declination or R.A. from measurements off the meridian 
could only be accomplished by a more devious method which calls for some 
knowledge of spherical trigonometry. This will now be explained. Figures 
traced out on the surface of a sphere are called spherical figures. Thus two 
parallels of latitude and two meridians of longitude enclose a spherical 
quadrilateral. The peculiarity of such figures is that all their dimensions are 
measured in fractions of the circumference of a circle, i.e. in degrees. If three 



Children of the Sun 



189 




FIG. 1 18. INTERSECTING FLAT PLANES ON WHICH THREE GREAT CIRCLES 
OF A SPHERE LIE 

This shows a globe in which three flat planes have been sliced through two meridians 
of longitude (along PA and PB), and through the equator (AB). Each of these planes 
cuts the surface of the terrestrial sphere in a complete circle, the centre of which is 
the centre of the sphere. Where they intersect on the surface they make the corners 
of a three-sided figure of which the sides are all arcs of great circles^ i.e. circles with the 
same centre and the same radius as the sphere itself. Such a figure is called a spherical 
triangle. It has three sides, PA, PB, and AB, which we shall call b (opposite B), a 
(opposite A), and p. It has also three angles B, A, and P (PBA, PAB, and APB). What 
you already know about a map will tell you how these angles are measured. The 
angle APB is simply the difference of longitude between the two points A and B 
marked on the equator, and it is measured by the inclination of the two planes which 
cut from pole to pole along the axis of the globe. You will notice therefore that, since 
the earth's axis is at right angles to the equator plane, the plane of AB is at right angles 
to the plane of PA and of PB; and since we measure angles where two great circles 
traced on a sphere cut one another by the angle between the planes on which the 
great circles themselves lie, the spherical angle PAB is a right angle, and so is PBA. 
Thus the three angles of the spherical triangle are together greater than two right 
angles, an important difference between spherical triangles and Euclid's triangles. 
In practice, of course, it is a lot of trouble to draw a figure like this. So we measure 
the angles in one of three other ways which only involve flat geometry, which we have 
already learnt. These are : 

(a) The geometry method : The angle BPA between the spherical sides PB and PA 
is the same as the flat angle RPQ between the tangents RP and QP which touch PB 
and PA at their common point, i.e. the "pole" P of the equatorial circle. 

(b) The geography method: Remembering that BPA is simply the number of 
degrees of longitude between A and B, you will see that it is simply the number of 
degrees in the arc cut off where the great circles on which PB and PA lie intersect any 
circle of latitude, i.e. any circle of which the plane is at right angles to the line joining the 
two poles where the great circles intersect above and below. 

(c) The astronomy method : This is illustrated in the next figure. 



190 



Science for the Citizen 



intersecting great circles, i.e. circles of which the centres coincide with that 
of a sphere, are traced out on its surface, the figure bounded by three of 
their circular arcs is called a spherical triangle. Its three sides (A, B, C) 
are measured in degrees, analogous to degrees of latitude along a meridian 
of longitude. Its three angles (a, 6, c) are measured like the angular differ- 
ence between meridians of longitude. The arc which measures the angle 
between them is the arc cut off by the two great circles from the equator 
where the sphere meets the plane drawn midway at right angles to the axis 
through the points where the great circles intersect. This is explained in 




FIG. 119 

The angle QES between the arcs QE and SE is the angle QOS between their inter- 
secting planes, and 

QOS = 90 - QOZ 

POZ = 90 - QOZ 

So the angle between two spherical arcs is the angle between the poles of the great circles 
on which they lie. 

Figs. 118 and 119. You will not find it difficult to visualize the meaning of 
a spherical triangle if you think about one of the most elementary problems 
of navigation, calculating the shortest course of a ship between two ports. 
The shortest course on the earth's spherical surface is the flattest arc 
which can be traced between two places. The flattest arc is the arc of the 
circle of largest radius, i.e. that of the sphere itself. Hence the shortest 
course is the arc of a great circle. This forms one side (Fig. 122) of a spherical 
triangle of which the other two arcs are two meridians of longitude also great 
circles. The length of these two arcs (Fig. 122) is known if the latitude of each 
port is known, and the angle between them is the difference of longitude. So 
finding a ship's course is finding the length of the third side of a spherical 



Children of the Sun 



191 




FIG. 120 

With the aid of the key shown in the next figure the two fundamental formulae for 

the solution of spherical triangles such as ABC in this one can be deduced from the 

formulae for solving flat triangles given in Fig. 56, Chapter II. 

P Q2 = PO 2 + QO 2 _ 2 po . QO cos a 

PQ 2 = PA 2 -f QA 2 - 2PA . QA cos A 

/. (PO 2 - PA 2 ) + (QO 2 - QA 2 ) - 2PO . QO cos a + 2PA . QA cos A - 
/. 2PO . QO cos's = 2AO 2 -f- 2PA . QA cos A 
Divide through by 2PO . QO 3 then 

AO AO , PA QA 



= cos POA cos QOA -f sin POA sin QOA cos A 
= cos b cos c + sin b sin c cos A 

The formula for getting the third side (a), when you know the other two (b and c) 
and the included angle A 3 is, therefore : 

cos a = cos b cos c -f sin b sin c cos A (i) 
cos A sin b sin c = cos b cos c cos a 

cos 2 A sin 2 b sin 2 c cos* b cos 2 c 2 cos a cos b cos c -f cos* a 
Now make the substitution cos 2 A = 1 sin 2 A> etc. 
(1 - sin 2 A) sin 2 b sin 2 c = (1 - sin 2 b) (1 - sin 2 c) 

2 cos a cos b cos c + (1 sin 1 a) 
.'. sin 2 b sin 2 c sin 2 A sin 2 b sin 2 c 1 sin 2 b sin 2 c 

-f- sin 2 b sin 2 c 2 cos a cos b cos c -f 1 sin* a 
After taking away sin 2 b sin s c from both sides this becomes 

sin 2 A sin 2 b sin 2 c 2 sin 8 a sin* b sin 2 c 2 cos a'cos b cos c 

Just by looking at this you can see that the right-hand side would be the same if 
we had started with 

cos c = cos a cos b -f sin a sin b sin C 
in which case we should have found 

sin 2 C sin 2 a sin 2 b = 2 sin 2 a sin 2 b sin 2 c 2 cos a cos b cos c 
Hence we can put 

sin 2 C sin*a sin 2 b = sin 2 A sin 2 b sin 2 c 
Dividing by sin 2 b y we get 

sin 2 C sin 2 a sin 2 A sin 2 c 
sin C sin a = sin A sin c 

. _ . sin A sine ,.. 
or sm C - : - (11) 



192 Science for the Citizen 

triangle if we know two others and the angle included between them. This 
can be done by using the first formula in Fig. 120. 

Flat trigonometry gives us rules for finding any of the other three dimen- 
sions (A, B, C, a, b, c) of an ordinary triangle, if we know one side (a, i, or c) 
and any two of the remaining dimensions (two other sides, two angles or 
an angle and a side). There are analogous rules for the solution of spherical 
triangles. The fundamental rule is 

cos a = cos b cos c + sin b sin c cos A (1) 

This is analogous to and derived from the flat triangle formula shown in 
Fig. 56: 

a~ =^ b 2 ~\- c 2 2bc cos A 
Fig. 56 also shows a second formula for flat triangles : 

a sin C c sin A 

sin A = or sin C = 

c a 

The corresponding formula for spherical triangles is 

^ sin c sin A 

sinC- : (2) 

sin a 

With the aid of a paper model as directed in Figs. 120 and 121 you will 
be able to overcome the difficulties of envisaging figures drawn on a sphere, 
and to see how the formulae for solution of spherical triangles follow from 
those for flat triangles. To apply them correctly you will need to recall some of 
the more elementary formulae in flat trigonometry* and the convention that 
angles measured east of a line of reference are negative. You will probably 
find it helpful to practice the use of the first formula by examples like the 
one shown in Fig. 122. Most atlases give the length of the ship's course 
between large ports, and you can therefore check your answer. The distances 
are usually given in sea miles (1 ' of the earth's circumference, i.e. a great 
circle of the terrestrial globe is 360 x 60 sea miles). 

To find the R.A. of a star off the meridian it is first necessary to find its 
declination with the cosine formula (1). The local position of every star at 
any instant can be placed at the corner of a spherical triangle (Fig. 123) 
like the Bristol-Kingston triangle of Fig. 122. One side (b) y like the polar 
distance of Kingston, is the arc between the celestial pole and the zenith 
along the prime meridian. The elevation of the pole is the latitude (L) of the 
observer. So b = 90 L. One side (c) 9 like the polar distance of Bristol, 
is the arc between the star and the zenith on its own great circle of azimuth. 
This arc is its zenith distance (c = z.d.). The angle A between its azimuth 
circle and the prime meridian which cuts it at the zenith is its azimuth 
(A = azim.). Between the ends of these two arcs passes the great circle of 

* sin A = cos (90 - A), cos A = sin (90 - A) 
sin A = sin (A) =* sin (180 A) 
cos A = cos (A) = cos (180 A) 
cos (180 A) = -cos A; sin (180 A) =- T sin A 



Children of the Sun 



193 




PQ 2 =PO 2 4-Q0 2 -ZPO,QO cos 

= PO a 4-Q0 2 -2PO,QO Cos a. 





PQ 




PA^4- QA 2 2 PA .QA COS 
=PA 2 -4- QA^- 2PA.QA COS A 




FlG. 121 

Paper model key to the geometrical relations of the previous figure (Fig. 120) 

G 



194 



Science for the Citizen 



right ascension which joins the star to the celestial pole, and the length of the 
arc between the star and the pole on its R.A. circle is its polar distance. Since 
the celestial pole is 90 from the celestial equator, the star's polar distance. 




FIG. 122 

The latitude of Bristol is 51 U 2<> 7 N. of the equator, and therefore 38 34' from the 
pole, along the great circle of longitude 2 35' W. The latitude of Kingston is 18 5', 
i.e. it is 71 55' from the pole along the great circle of longitude 76 58' W. The arc 
joining the pole to Bristol (c\ the arc joining the pole to Kingston (ft), and the arc (a) 
of the great circle representing the course from Bristol to Kingston form a spherical 
triangle, of which we know two sides (b and c), and the included angle A, which is 
the difference of longitude 76 58' 2 35' ~ 74 23' between the two places. So 
we can find a from the formula (i) in Fig. 120 by putting 

cos a = cos 71 55' cos 38 34' + sin 71 55' sin 38 34' cos 74 23' 
From the tables : 



cos a = 0-3104 
= 0-4022 



X 0-7819 -f 0-9506 x 0-6234 x 0-2692 



Thus a is approximately 66 of a great circle, i.e. a circle of the earth's complete 
circumference. The length of one degree of the earth's circumference is approximately 
69 miles. So the distance is approximately 

66 J x 69 = 4,577 land miles (3,980 sea miles) 

which is the third side of our spherical triangle, is the difference between one 
right angle and its declination (a = 90 declin.). Applying the formula 
we have 

cos (90 declin.) cos (90 lat.) cos z.d. 

+ sin (90 lat.) sin z.d. cos azim. 



Children of the Sun 195 

This may be written 

sin declin. sin lat. cos z.d. + cos lat. sin z.d. cos azim. 

In applying this you have to remember that we have reckoned azimuth 
westward from the north point. The azimuth is reckoned positive west and 
negative east of the north point. If the star transits south of the zenith, the 
azimuth A reckoned west or east of the south point is equivalent to 180 ^ A 
from the north point. Since cos (180 i A) = cos A, cos (azim.) is always 



(Pole) 




FIG. 123. THE STAR TRIANGLE 

of the opposite sign, if reckoned from the south point, and the formula 
becomes : 

sin declin. sin lat. cos z.d. cos lat. sin z.d. cos azim.* 

This means that if you know the azimuth and zenith distance of a heavenly 
body at one and the same time, you can calculate its declination without 
waiting for it to reach the meridian. You cannot get the latitude of the place 
from an observation on a star of known declination directly by using the same 
formula; but if you have the z.d. and azimuth of any two stars taken at one 

* When a star is crossing the meridian its azimuth is zero or 180 and since cos 

1 and cos 180 1, sin declin. = sin lat. cos z.d. cos lat. sin z.d. 
Since sin (A B) = sin A cos B cos A sin B 

sin declin. sin (lat. z.d.) 
This is the formula given in Fig. 41 when no sign is attached to the z.d. 



196 Science for the Citizen 

and the same place, you can calculate its latitude provided you have an 
almanac or star map to give the declination of the stars. In general the 
arithmetic takes less time than waiting about for one bright and easily 
recognized star to cross the meridian. 

Another application of the formula just derived gives the direction of a 
heavenly body when rising or setting at a known latitude, or conversely 
the latitude from the rising or setting of a star. At the instant when a heavenly 
body is rising or setting its zenith distance is 90. Since cos 90 = and 
sin 90 = 1, the formula then becomes 

sin declin. = cos lat. cos azim. 
On the equinoxes when the sun's declination is 0, 

cos lat. cos azim. = 
.*. cos azim. =* 
/. azim. = 90 

That is to say, the sun rises due east and sets due west in all parts of the world 
that day. To find the direction of the rising or setting sun at latitude 51 1 N. 
(London) on June 21st, when the sun's declination is 234 N., we have only 
to put 

sin 23| = cos 51| cos azim. 
From the tables, therefore: 

0-3987 = 0-6225 cos azim 
/. cos azim. = 0-6405 
azim. = 50J 

Thus the sun rises and sets 50J from the meridian on the north side, or 
90 50 J = 39f north of the east or west point. Conversely, of course, you 
can use the observed direction of rising and setting to get your latitude. 

If you look at the star triangle shown in Fig. 123, you will see that the 
angle C between the arc which represents the star's polar distance and the 
arc by which is the angle (90 lat.) between the observer and the earth's pole, 
is the angle through which the star has rotated since it was last on the meridian. 
Since the celestial sphere appears to rotate through 360 in 24 hours, i.e. 15 
an hour, this angle C is sometimes called the hour angle of the star, because 
you can get the time (in hours) which has elapsed since the star made its 
transit by dividing the number of degrees by 15. If you know when the 
star crossed the meridian by local time, you also know how long has 
elapsed since the sun crossed the meridian because time is reckoned that 
way, and if you know the sun's R.A. on the same day, you know how long 
has elapsed since T crossed the meridian. Thus all you have to do to get 
the star's R.A. is to add the sun's R.A. to the star's time of transit. 

So to get the star's R.A. from its altitude and azimuth at any observed 
time, we need to determine one of the other angles of a spherical triangle of 
which we already know two sides and the angle between them. This is done 
by the second formula which tells us that : 



Children of the Sun 197 

sin A sin c 



sin C = 



sin a 



In our original triangle of Fig. 123, A is the azimuth, c is the zenith distance, 
and a the polar distance (90 declin.) of the star, i.e. sin a = cos declin. 
Hence 

sin azim. sin z.d. 

sin hour angle = 

cos declin. 

As stated already, the azimuth is reckoned positive west and negative east 
of the north point. Since sin ( A) = sin A, the sine of the azimuth is 
negative if the star has not yet reached the meridian. If it transits south of 
the zenith its azimuth A east of the south point is equivalent to (180 -f A) 
measured west of the north point. Since sin (180 + A) = sin A, the sine 
of the azimuth is also negative if measured east of the south point. 

Suppose that the star Betelgeuse in Orion is found to have the hour angle 
10 when it is west of the meridian at 8.40 p.m. local time. It crossed the 
meridian xf hour = 40 minutes before, i.e. at exactly eight o'clock, and its 
R.A. is greater than that of the sun by 8 hours. If the sun's R.A. on that day 
were 21 hours 50 minutes, the sun would transit 2 hours 10 minutes before T, 
i.e. T would transit at 2.10 p.m., and Betelgeuse 8 hours minutes 2 hours 
10 minutes = 5 hours 50 minutes after T. So its R.A. would be 5 hours 
50 minutes. 

The same formula also tells you how to calculate the time of rising and 
setting of stars in any particular latitude. At rising or setting the z.d. of a 
heavenly body is 90, and sin 90 = 1. So the formula becomes 

sin azim. 
sin hour angle -= 



cos declin. 

The azimuth of a rising or setting star can be found from the formula already 
given, i.e. 

sin declin. 

cos azim . = - : - 
cos lat. 

As an example we may take the time of sunrise on the winter solstice in 
London (Lat. 51). By the last formula the azimuth of the rising and setting 
sun is 50 from the south point on the winter solstice. So at sunset 

. . . 

Smhourangle== 



C os(-23F) 

0-7679 
~~ 0-9171 
= 0-8373 

For sunrise the azimuth will be east, therefore of negative sign, and the 
result is 0-8373. Since 0-8373 is the sine of 56 51', the time which elapses 
between setting or rising and meridian transit (i.e. noon, since it is the sun 
with which we are dealing) is (56| -f- 15) hours, i.e. 3 hours 47 minutes. Thus 



198 Science for the Citizen 

sunrise would occur at 8.13 a.m., and sunset at 3.47 p.m. Daylight lasts 
roughly 7-|~ hours. This calculation differs by about 6 minutes from the value 
given in Whitaker. This is partly due to approximations made in the arith- 
metic, and partly due to other things about which you need not worry, 
because you will not find it difficult to put in the refinements when you 
understand the basic principles. 

The same formula would apply to calculating the times of sunset or sunrise 
on June 21st, when the lengths of day and night are reversed. Since sin C 
= sin (180 C), 0-8373 may be either sin 56 51' or sin (180 56 51'), 
i.e. sin (123 9'). Fig. 124 shows you at once which value to take. An equa- 
torial star rising due east passes through 90 in reaching the meridian. A star 
south of the equator passes through a smaller and a star north of the equator 
through a larger arc. So if the declination of a heavenly body is north (like the 
sun on June 21s/), we take the solution as sin (180 C), and if south as sin C. 
Thus the hour angle of sunrise and sunset on June 21st would be (123 \ -f- 15) 
hours = 8 hours 13 minutes, i.e. sunrise would be at 3.47 a.m. and sunset 
at 8.13 p.m. solar time.* 

The following data, determined by a home-made instrument like the one 
shown in Fig. 115, illustrate how you can find the position of any star on 
the star map or make your own star map from observations off the meridian. 
At 9.5 p.m. (G.M.T.) near Exeter (Lat. 51 N. Long. 3J W.) the bright 
star Procyon in Canis Minor was seen on February 10th at 49 below the 
zenith and 28| east of the south point. The star transits south of the zenith, 
so we use the difference formula. From tables of sines and cosines we get 

sin declin. = sin 51 cos 49 cos 51 sin 49 cos 284 

= 0-7771 x 0-6561 0-6293 X 0-7547 x 0-8788 
= 0-0925 
/. declination = 5 18' 

Since the star is east of the meridian its hour angle is negative, and 

sin 49 sin 28 J 



sin (hour angle) 



cos (5 18') 
0-7547 x 0-4772 



0-9957 
Hour angle =- - 21i 

In time units 21^ is 1 hour 25 minutes, and since the sign is negative, this 
means that the star will transit 1 hour 25 minutes later. On February 10th 
(39 days before March 21st) the sun's R.A. is about 21 hours 35 minutes. 
Under "equation of time" Whitaker states that we must add 14^ minutes to 
apparent (sundial) time to get mean time. Hence the time of observation was 
(9 hours 5 minutes ~- hours 14J minutes) = 8 hours 50 minutes Green- 
wich sundial time. Since Exeter is 3 W., the Exeter time is 13 minutes slow 
by Greenwich (i.e. it is 11.47 a.m. at Exeter when it is noon at Greenwich). 

* The remarks here made apply to the northern hemisphere 3 where the bulk of the 
world's population lives at present. Australians and New Zealanders will not need to 
be told how to make the necessary adjustments. 



Children of the Sun 



199 



r 



Hour angle of 




April 15 

>f% X , , 

vSun s //our a^ 



4- 7 



FIG. 124. RELATION BETWEEN R.A., HOUR ANGLE OF A STAR, AND LOCAL TIME 

About April 35th and August 28th the sun's declination is a little less than 10 N. and 
its hour angle of rising and setting is 103. The shaded area is the arc of the sun's path 
below the horizon on the earlier date when its R.A. is 1 hour 30 minutes. The time is 
(April 15th) 2 a.m. (14 p.m.). So the sun's hour angle is -f 14 hours or 10 hours. 
The sun will transit 10 hours hence; nr will transit 10 hours 1 hour 30 minutes or 
x hours 30 minutes hence. Vega's R.A. is 18 hours 35 minutes, hence it transits 18 hours 
35 minutes after <v and 5 hours 25 minutes before <v . Therefore Vega will transit 
8 hours 30 minutes 5 hours 25 minutes or 3 hours 5 minutes hence, and the hour 
angle of Vega must be 3 hours 5 minutes or 4 20 hours 55 minutes. The formula 
for the star's R.A. if its hour angle at a given instant of solar time is known is therefore 
derived as follows: 

sun's hour angle star's hour angle = star's R.A. -- sun's R.A. 

Solar time star's hour angle = star's R.A. sun's R.A. 

star's R.A. ~ surfs R.A. 4- solar time star's hour angle. 

Thus., if Vega's hour angle is + 20 hours 55 minutes or 3 hours 5 minutes at 1 4 
p.m. when the sun's R.A. is 1 hour 30 minutes 

R.A. of Vega = 1 hour 30 minutes + 14 hours minutes ( 3 hours 5 minutes) 
--18 hours 35 minutes 

Or using positive quantities only 

= 1 hour 30 minutes + 14 hours minutes 20 hours 55 minutes 
s= 5 hours 25 minutes = -}- 18 hours 35 minutes 



200 



Science for the Citizen 



So the time of observation by the Exeter sundial was really (8 hours 
minutes hours 13 minutes) = 8 hours 37| p.m. The star then had still 
to rotate 1 hour 25 minutes before transit. So transit would occur at (8 hours 
38f minutes + 1 hour 25 minutes) = 10 hours 2| minutes p.m., i.e. 10 
hours 2-| minutes after the sun's transit. The sun on that day transits 21 hours 
35 minutes after T 5 and hence (24 hours minutes 21 hours 35 minutes) 
= 2 hours 25 minutes before T, i.e. T transits at 2.25 p.m. on that day. 




FIG. 125 
In Ptolemy's system four classes of motion were recognized : 

(i) The diurnal rotation of the whole heavenly sphere from east to west. In this 
motion the stars maintain a constant position relative to one another. A heavenly body 
(moon, star, or planet) west of the sun rises before sunrise. A heavenly body east of 
the sun sets after sunset. In the position here drawn, Venus and Mars are seen as 
"morning stars," Mercury and Jupiter as "evening stars." 

(ii) The daily increase of the moon and sun in R.A. as they retreat from west to east. 

(iii) Two independent motions of each planet, one the epicyclic orbit of the planet 
about an imaginary fixed point, the other the deferent orbit of this fixed point around 
the earth as world centre. The deferent motion was analogous to the daily increase of 
the sun's or moon's R.A. The epicyclic motion might have, e.g. when Mars is at 
(a), the same direction, or, e.g. when Mars is at (), the opposite direction to the 
deferent motion. Hence planets may have alternate periods in which the R.A. increases 
daily and slowly diminishes. 

Hence the star's transit was (10 hours 2| minutes 2 hours 25 minutes) 
=7 hours 38 minutes after T, i.e. the R.A. of Procyon is 7 hours 37| minutes 
if the observations were correct. With the best instruments the value given is 
7 hours 36 minutes. 

THE ALEXANDRIAN VIEW OF PLANETARY MOTION 

When these methods are applied to the observation of the position of the 
planets day by day over a period of years, the latter are all found to be alike 



Children of the Sun 



201 



in one respect. Each planet exhibits long periods in which its R.A. increases 
steadily like the R.A. of the sun or moon, alternating with shorter periods 
in which its R.A. diminishes at a slower rate, so that it appears to pause 
and double on its course at regular intervals. The times when a superior 
planet* like Mars transits near midnight and is visible throughout the night 
correspond to the times when its apparent motion is retrograde, or in the 
opposite direction to the annual motion of the sun and the monthly motion 
of the moon among the fixed stars. At the time when the motion of such a 




FIG. 1:26 

In the Copernican system only three classes of motion need be recognized : 

(a) the earth's diurnal motion about its own axis from west to east, (b) the moon's 
orbital motion from west to east, (c) the orbital motion of all planets (including the 
earth) from west to east. In the positions here shown Mars and Jupiter are in opposi- 
tion, i.e. on the meridian at midnight. Mercury is at its extreme easterly position 
(maximum elongation) as an evening star. Venus is at its extreme westerly position 
(maximum elongation) as a morning star. Neither of them can ever be seen on the 
meridian after dark. 

planet is most rapidly direct it is invisible. That is to say, it rises and sets 
about the same time as the sun. Thus there is a very close connexion between 
the apparent movements of the planets and their position relative to the sun, 
as we see them. This connexion was recognized by the Egyptians, who 
believed that the whole celestial sphere, including the sun, rotated around the 
earth as the centre, and that the planets rotated round the sun as the moon 
rotates around the earth. They placed the orbits of Mercury and Venus cor- 

* I.e. a planet whose orbit lies outside that of the earth. 
G* 



2O2 Science for the Citizen 

rectly between the sun and the earth. This explained why Mercury and Venus 
can never be seen throughout the whole night, since they can never be above 
the side of the earth opposite to that which is illuminated by the sun's rays. 
The theory which Hipparchus, and later Ptolemy, took from Apollonius 
was a decidedly backward step. Each planet (Fig. 125) moved around an 
imaginary centre in an orbit called its epicycle. Each imaginary centre was 
placed at the end of an imaginary or deferent spoke rotating round the earth 
itself. While accounting for the retrograde and direct motion of the planets, 
the theory of epicycles could recognize no significance in the connexion 
between the different phases of a planet's motion and its position relative to 
that of the sun. 

Its great defect was that it made the geometry of the heavens a good deal 
more complicated than the alternative view that the planets revolve around 
the sun; and the more accurate observations, which had been accumulated 
by the Arabian astronomers in the period which followed, made increasing 
demands upon mathematical ingenuity, as new epicycles were added to 
accommodate the theory with the facts. Copernicus, as we have seen, began 
his work when the forecasting of planetary occupations was becoming a 
matter of practical moment, and the possibilities of further improvement 
on the Ptolemaic system had been exhausted. There remained the alternative 
of starting from fresh assumptions. Copernicus went back to the doctrine 
of Aristarchus and put the sun at the centre of the whole planetary system, 
including the earth as a planet (Fig. 126). Having no telescopic information 
to reveal their different sizes as seen at different phases, he stuck to the 
idealistic belief that each planet moves in the most perfect plane figure, the 
circle. This assumption is so nearly true of Venus and Mars that it does 
not involve very serious inaccuracies, and makes it easier to understand 
how the position of a planet is calculated. So we may here suppose that the 
orbits of Venus and Mars are circles. 



THE HYPOTHESIS OF COPERNICUS 

The hypothesis which Copernicus adopted may be summarized under four 
headings : 

(1) The apparent diurnal rotation of the celestial sphere is due to the 

complete rotation of the earth about its polar axis in a period of 
24 hours. 

(2) The moon revolves around the earth in a period of 27| days. 

(3) The earth and the planets revolve in circular orbits about the sun in 

the same direction as the earth's diurnal motion. 

(4) The orbits of Mercury and Venus lie between the sun and that of the 

earth, while the orbits of Mars, Jupiter, and Saturn, lie beyond the 
earth's orbit. 

The tracks of the planets lie close to the ecliptic. So it is better to calculate 
their positions in celestial longitude and latitude (see p. 220) as Copernicus 
did. For the purpose of grasping the principles employed in tracing out their 
orbits it will be sufficient for our purpose if we use right ascension to measure 



Children of the Sun 



203 



their angular displacements. This is equivalent to projecting their movements 
on to the plane of the celestial equator. 
The first thing to notice (Fig. 127) is that it does not make any difference 



Mar. 21 
Sun enters ATI 



Dec 2J ( / 

V3-] 

Sun enLirs 



^--tf 



11 



Sept 23 



June 21 

Sun enters 

Cancer 



Sept. 23 
5o/i ent&rf L ibra. 




_ 

i 6 9 c 
o 



u/z. 
Capricorn 



March 21 _ 
5un outers Aries 

Copernlz&ru, 
FIG. 127 

Since the earth and sun are very close relative to the distance of the fixed stars (i.e. 
the annual parallax of a star is very small) the direction of the stars may be measured 
either from the earth as centre (Ptolemaic) or the sun as centre (Copernican) without 
making much difference. 

to the measurement of the sun's R.A. whether we put the earth or the sun 
at the centre of the star map. To calculate the position of a planet according 
to the Copernican hypothesis we need to know two things, (a) its distance 



204 



Science for the Citizen 



relative to the sun; (i) its sidereal period (P), i.e. the time which it takes to 
go round the sun. In the case of the inferior planet Venus, all we need to 
know is (i) the greatest elongation, i.e. the greatest difference between the 
R.A. of Venus and of the sun in the course of a Venus cycle; (ii) the length 
of the synodic period or Venus cycle, which has been explained already (see 
p. 182). The greatest elongation happens, of course, when the planet is farthest 
east or west of the sun. In other words, when the interval between its setting 
and sunset (or its rising and daybreak) is longest. A glance at Fig. 128 shows 




FIG. 128. MAXIMUM ELONGATION OF AN INFERIOR PLANET 



The R.A. of Venus is the angle v EV and the R.A. of the sun is TES. The difference 
or elongation is SEV. This is evidently greatest when the line (EV) joining the earth 
to the planet just grazes the latter 's orbit, i.e. when EV is the tangent to the circle of 
radius SV, and therefore at right angles to SV. In a certain year, when Venus was at 
that time an evening star, the elongation was greatest during the first week in July. 
The sun's R.A. was then 6 hours 55 minutes or 14 in Cancer and the R.A. of Venus 
was 10 hours 1 minute or 30 off the first point of Libra (^). The elongation SEV 
was therefore (150 104) 40. The radius of the earth's orbit is SE, that of the 
orbit of Venus is SV and since the triangle SVE is a right angled triangle 

~ = sin 46 = 0-72 
Thus the ratio of the orbits of Venus and the earth are as 72 : 100. 

you that this is when the angle between the sun, Venus, and the earth (SVE) 
is 90. The same figure (Fig. 128) shows that the angle between the sun, the 
earth and Venus (SEV) is the difference between the sun's R.A. and that 
of Venus, i.e. the elongation of the planet. So at maximum elongation the 
sun, the earth and Venus form a right-angled triangle in which 

SV 

sin SEV = 
oJti 



Children of the Sun 



205 



i.e. 



sin (maximum elongation) 



radius of the orbit of Venus 
radius of the earth's orbit 



The greatest possible angle between the sun and Venus is roughly 46.* 
Since sin 46 = 0-72, the ratio of the earth's distance from the sun to that 



Eartk 

'./7J935 




FIG. 129. THE VENUS CYCLE 

On March 12, 1934, the sun's R.A. was 352, being then 39 greater than that of 
Venus which was thus a morning star, rising before the sun. On October 17, 1935, 
584 days later the sun's R.A. was 202, being again 39 in excess of the R.A. of Venus. 

of Venus is 100 : 72. Hence you can take the first step in drawing a scale 
map for showing the relative positions of the earth and Venus by describing 
two circles of radii in the ratio 100 : 72 with the sun as centre. 
According to the hypothesis we are now adopting, the earth moves round 

* This is not exactly the same as the greatest difference in R.A. For simplicity we 
here neglect the fact that Venus and the sun do not usually have the same declination. 



206 Science for the Citizen 

*{f)( \ Q AA 

the sun through - ~ degrees per day. Suppose Venus moves through ^r- 

degrees per day. This means that Venus takes V days to go round the sun. 

T t_ f /^ 36 36oN \ 

It therefore gains ( - J degrees per day. We have seen (p. 1 83) that 

the Venus cycle, i.e. its synodic period or the time taken for the earth, the sun 
and Venus to regain, the same relative positions, is 584 days. Thus Venus 

360 
gains 360 in 584 days or degrees per day, so that 

*)o~t 

360 360 360 
684 = ~V" ~~ 365 



__ _ ___ _ 

V ~" 584 + 365 ^ 2920 
V = 225 days 

Thus the sidereal period of Venus is 225 days. In other words, Venus rotates 

360 
through ^p degrees per day. You can now see whether the hypothesis is 

satisfactory by observing the R.A. of Venus on any particular day (Fig. 130) 
and calculating what it will be at some later date. Seventy-five days 
later Venus will have moved through 120 and the earth through 74". 
According to Whitaker on September 23, 1934, when the sun's R.A. is 
180 (in degrees), the R.A. of Venus was 11 hours 8 minutes or 167, i.e. 
Venus was 13 west of the sun and just about at the end of its period 
as a morning star. Seventy-five days later, December 7th, Venus would 

75 
have revolved through -~ x 360 120 and the earth would have 

rjfi ~- Jf 

revolved through x 360= 74: i.e. the sun's R.A. has increased by 

74, and is now 254. If you now draw Venus and the earth in their new 
positions on your scale map, you will find that the R.A. of Venus is now a 
little greater than that of the sun. Venus is beginning to be an evening star. 
The angle between Venus, the earth and the sun (the elongation of Venus) 
is 5. According to Whitaker, the difference between the R.A. of Venus and 
the sun on December 7th was hours 19 minutes 37 seconds, or 5. 

The case of a superior planet like Mars can be dealt with in this way 
(Fig. 131). First find the length of the Mars cycle, i.e. the synodic period which 
elapses between two successive occasions when Mars, the sun, and the earth 
occupy the same relative positions. This is easily done by noting when 
Mars is in opposition, i.e. when it is on the meridian at midnight, and count- 
ing the number of days which intervene before its next midnight meridian 
transit. If you refer to Whitaker., you will see that Mars was in opposition 
(midnight meridian transit) on March 1 , 1933, and it was next in opposition 
on April 6, 1935, 766 days later. This makes the Mars cycle 766 days. So, if P 
is the length of the sidereal period 



Children of the Sun 



207 



j. 
p 



i 

365 



1 

766 



1 
697 



The earth gets back to its original position (Ej) after 365} x 2 = 730| days. 
To the nearest day this is 34 days after Mars completes a sidereal period of 
697 days. At the end of a sidereal period of Mars, the earth is therefore 
34 west of the position which it occupied at the beginning. If we now 
find the R.A. of Mars,, we know its elongation, the angle SE 2 B 5 as in Fig. 131 . 




Venus" 



FIG. 130. CHECKING THE COPERNICAN HYPOTHESIS 

Since the sidereal period of Venus is 225 days, Venus goes through 120 in 75 days, 
the interval between September 23rd and December 7th. On September 23, 1934, 
the R.A. of Venus was 11 hours 8 minutes or 13 less than that of the sun (12 hours 
or 180). On December 7th the R.A. of Venus, which has advanced through 120 of 
its orbit> will be found to be 4i in excess of the sun (2534). Thus its R.A. is 258 or 
17 hours 12 minutes. 

Since Mars was in opposition when the earth was at E 15 it was then situated 
somewhere on the line SEjA. After 697 days it is somewhere on the line 
E 2 B. According to the Copernican view, it is presumably in the same place 
once more. So it must be at M where these two lines intersect. Hence 
we can now put the orbit of Mars on the scale map by drawing a circle of 
radius SM. The radius of its orbit is then found to be 1 52 times that of the 
earth's orbit. 

The reasoning given above and the figure based on it (Fig. 131) need 
qualifying. The synodic period is not absolutely constant. The average length 



208 



Science for the Citizen 



of the Mars cycle is nearly 780 days, and the sidereal period is therefore 687 
days. Hypothesis and observation can be tested in the same way for Mars 
and Venus. The only thing which remains to be explained is the retrograde 
movement which the planets show. As stated, the R.A. of a superior planet 
like Mars diminishes daily when it is visible during the greater part of the 
night, i.e. about the time when it is in opposition. Fig. 132 shows you how 




IG. 131. THE ORBIT OF A SUPERIOR PLANET 

About March 1, 1933 a Mars is in opposition, the sun being roughly 20 from the 
First Point of Aries. So it lies somewhere on the line SEjA. By January 27, 1935, 
Mars has returned to the same position in its orbit. The sun's R.A. is now about 64 
west of T and the R.A. of Mars is 13 hours 15 minutes, i.e. it is 19 in Libra. It 
now lies somewhere on the line E 2 B. The radius of the circle drawn through M 
where EjB and E 2 B cut is 1-52 times SE 3 or SE 2 , the radius of the earth's orbit. 
Hence Mars is one and a half times as far away from the sun as the earth is. 

this happens. The sidereal period of Mars itself is a little less than twice 
that of the earth (687 : 365). For simplicity, consider an imaginary planet M 
which revolves like Mars in an orbit 1| times the width of the earth's orbit, 
taking exactly twice as long as the earth to make a complete revolution. 
So, if the earth goes through 40, the imaginary planet goes through 20 in 
the same time. On the left, the earth is shown in two positions at the begin- 
ning of March and the middle of April. In the first it is approaching, in the 
second at opposition. You will see that the R.A. of the planet changes from 
180 + a (a off the first point of Libra) to 180 + b. Since b is manifestly 



Children of the Sun 



209 



smaller than a, the R.A. of the planet is diminishing daily at this stage in 
its course. On the right, the planet is approaching conjunction. It is just 
seen after sunset at M tt and is totally obscured by the sun when it has moved 
on through 20 to M ft . Meanwhile the earth moves from E a to E 6 through 40. 
At M a the planet is in Pisces, at M & in Aries, and its R.A. is therefore in- 
creasing daily. 

Before dealing with the imperfections of the Copernican hypothesis one 
result may be pointed out. Early estimates of the sun's distance like that of 
Hipparchus were very inaccurate. Owing to its great distance the sun's 



"A- 





FIG. 132. RETROGRADE AND DIRECT MOTION OF A PLANET 

The outer planet M takes twice as long to traverse its orbit as the inner one E. Hence 
when E goes through 40, M rotates through 20. Their relative motions resemble 
what a passenger in one train would experience if travelling in a circular track con- 
centric with another one on which a train was also moving. When E and M are close 
they are travelling in the same direction and E is gaining on M. The motion of M is 
then retrograde, i.e. the R.A. of M is decreasing. When M and E are in opposite side 
of their orbits they are travelling in opposite directions and both motions make the 
R.A. increase. So the motion is direct. 

parallax is only about 9" or 400 of a degree. The Copernican hypothesis 
gives us a simple way of estimating the sun's distance without recourse to 
the direct measurement of such a small quantity. Having now made a scale 
map (Fig. 133) of the orbits of the planets by combining Fig. 128 with 
Fig. 131 and others like them, we can at once deduce the distance between 
the sun and any planet if we can find the actual distance between any points 
on our scale. Thus we can use the parallax of the nearest planet to get the 
sun's parallax. For instance, the distance between the earth and Venus when 

100 72 

the latter crosses the sun's disc is r times or only about a quarter of the 

luu 



2IO 



Science for the Citizen 



distance between the earth and the sun. A minor planet Eros discovered in 
the latter half of the nineteenth century comes within a distance equivalent 
to a sixth of the radius of the earth's orbit. By determination of the parallaxes 
of near planets like Eros, Venus, and Mars, we know the sun's parallax with 
great accuracy and the average distance deduced is about 93 million miles. 
This estimate agrees very closely with two other estimates based on the 
optical phenomena of aberration and line spectra (see Chapter VI). A fairly 
close approximation to the diameter of the sun can be got without telescopic 
equipment. Since the moon's disc just covers the sun in a total eclipse, the 
sun's angular diameter is very nearly the same as that of the moon. That is 




FIG. 133. SCALE DIAGRAM OF THE ORBITS OF THE FOUR INNER MAJOR PLANETS 

to say, it is roughly half a degree. So the sun's diameter calculated in the 
same way as the moon's is between three-quarters of a million and one 
million miles. 



THE IMPERFECTIONS OF THE COPERNICAN HYPOTHESIS 

Calculations based on the Copernican assumption that the orbits of the 
planets are circular do not yield conclusions which are sufficiently accurate. 
Detectable errors were recognized before the advent of the telescope and all 
the refinements of measurement which followed it. An immense array of 
new data about the planet Mars collected by Tycho Brahe in the latter half 
of the sixteenth century made it possible to analyse the movement of Mars 
more thoroughly than Copernicus or his predecessors had done. If, instead 
of determining the radius of the orbit by drawing a circle through the point 
M where E X A and E 2 B intersect in Fig. 131, we note the successive positions 
of the earth after several complete sidereal cycles beginning on different dates, 



211 




FIG. 134. THE ELLIPSE 

Since the loop is of fixed length, the sum of the distances a and b of any point from 
the two foci is the same. Half the sum is d in the diagram, i.e. a -f b 2d 

The ellipse is symmetrical about two unequal diameters. One, called the minor axis., 
at right angles to the line between the foci, joins the opposite points where a = b d. 
If m is half the minor axis Fig. (c) shows you that 



(i) 



The other, called the major axis, is a continuation of the line joining the two foci. 
You will see from the diagram, in which half the major axis is called M, that 

M = a + c and M = b c 

:. 2M - a -f b - *2d 

M = d (!!"> 



We call the ratio of c, the distance between the centre and the focus, to d y the distance 
along the major axis from the centre to the boundary, the eccentricity of the ellipse, i.e . 



Combining (ii) and (iii) 



= - or 



2 - M 2 (1 



c de 



(iii) 



212 Science for the Citizen 

we can map out its entire course by the method illustrated in Fig. 135, which 
actually refers to the planet Mercury. An examination of all Tycho Brahe's 
data led Kepler to the conclusion that Mars does not move in a circle with 
the sun as centre. The orbit is an ellipse with the sun at one focus. 

To understand Kepler's laws we must be acquainted with some simple 
properties of the ellipse. An ellipse is the figure which can be drawn with a 
pencil, two pins and a piece of cotton as in Fig. 134. The position of the 
two pins represent the two "foci" of the ellipse. If the two foci of the ellipse 



Sept. 25 
Sept.Q 




FIG. 135. THE ORBIT OF MERCURY 

are very close together it becomes undistinguishable from a circle. The 
connexion between the circle and the ellipse is brought out more precisely 
by the index known as eccentricity (e). The broadest diameter of an ellipse 
is called its major axis. The narrowest is called the minor axis. The two foci 
lie on the major axis. If we measure along the major axis the distance of either 
focus from the boundary in both directions, the ratio of the difference between 
the greater and smaller distance to the length of the major axis is called the 
eccentricity of the ellipse. If its eccentricity is 0, the two foci coincide and 
an ellipse becomes a true circle. The geometry of the ellipse (Figs. 134 and 
136) shows that e can also be defined as : 



minor axis 



ymajor axisy 



Children of the Sun 



213 



So if we say that the eccentricity of the orbit of Mercury is 0*2, we mean 
that 



(0-2) 2 



1 - 



/minor axis\ 2 
y major axisy 



i.e. the minor axis and major axis are in the ratio 49 : 50. Since the eccen- 
tricity of the orbit of Mercury is more than twice that of any of the other 
major planets (except the newly discovered planet Pluto) it is clear that the 



Minor 
(y axis 



X 



FIG. 136. THE EQUATION OF THE ELLIPSE 

a 2 - y + (x + cy - y -f x 2 -f 2cx -f c 2 . . . . (iv) 
fc 2 =- y 4- (* - c) 2 - y -f Jc 2 - 2cx 4- f a . . . . (v) 

Adding the left-hand side of (iv) to the left-hand side of (v) and the right-hand side 

of (iv) to the right-hand side of (v), we get 

a 2 -f fc 2 = 2;y; 2 -f 2;c 2 4- 2c 2 

2^ 2 -f 2x~ + 2^ 2 e 2 (vi) 

Subtract (v) from (iv) : 



Since 



Take (vii) from (vi) : 
Add (viii) and (vi) : 

But (Fig. 135) 



From (ii) and (iii) 



^ 2 - fc 2 = (a + V) (a - b) 
:. (a + b) (a - b) - 4dex 
2d (a -b) = 4dex 
a -b ^2ex 
.'. (a - b^ = le*x~ 
or a 1 - 2a6 + b* - 4e 2 * 2 



2ab = 2y- - 

(a -f b) 2 = 
(a 



(vii) 
(viii) 



_ rfv = ,2 



3,2 



d* 



M 2 



214 



Science for the Citizen 



Copernican doctrine of circular orbits is not very wide of the mark. The 
eccentricities of the orbits of Venus and Mars are respectively 0-007 and 
0-093. So the orbit of Venus is very nearly a perfect circle. 

In his first study on Mars, Kepler had to work on the assumption that the 
earth's orbit is practically circular. By good fortune it happens to be so. The 
eccentricity of the earth's orbit is only 0-017. So the orbit of Mars is deci- 
dedly more flattened. With the aid of Figs. 134 and 136, which give the 
Cartesian equation of the ellipse, you will be able to see how Kepler's first 
law can be tested. Having made a graph of the actual positions, measure 
the greatest width (major axis) and the perpendicular width through the 




V Dec. 37 
/ (perig&ej 



March 21 

FIG. 137. PLOTTING THE EARTH'S ORBIT 
The dates of perigee and apogee are Kepler's. They are now about 3 days later. 

midpoint of the longest diameter. This will be the minor axis. Fig. 136 
shows you that,, if an ellipse is drawn with the minor axis lying along the y 
and the major axis along the x reference lines, 



=. 

m^ M 2 

From this you can tabulate corresponding values of x and y , since m and M 
are known. If Kepler's first law is a good one, the points deduced from 
the equation should lie closely in the same curve as those based on direct 
observation. Having satisfied himself that the figure was not a true circle, 
Kepler explored nineteen hypotheses before he found one which was entirely 
satisfactory. 
The publication of Kepler's analysis of the orbit of Mars was immediately 



Children of the Sun 215 

followed by the invention of the telescope. With new optical equipment it 
was now possible to measure smaller angles. With refined measurement the 
angular diameter of the sun is seen to vary appreciably in the course of a 
year. That is to say., the earth is nearer to the sun at some times than at 
others. The distance at any time is inversely proportional to the apparent 
size. So if we make a date circle along the radii of which the earth lies at 
different times of the year, as in Fig. 137, we can measure off distances 
proportional to the actual distance of the sun on any day. If we first measure 
off SA as the earth's distance on December 31st (when the earth is nearest 
to the sun), and find that on July 1st the sun's angular diameter is x times as 
large on December 31st as on July 1st, the earth's distance SB on July 1st, 
is SA -f- x. In this way we can plot out the earth's orbit from day to day. 
Although it is very nearly circular far more so than the accompanying 
figure would suggest it is easy to detect that the earth does not move in a 
circle with the sun as centre. If you draw a series of ellipses of different 
eccentricities by the method shown in Fig. 135, you will find that the foci 
can be relatively far apart without producing a very noticeable departure 
from the shape of a circle. If the foci are far apart either of them must 
be decidedly nearer to one end of the major axis than to the other. Measure- 
ment of the form of the earth's orbit shows that the sun is at one focus of 
an ellipse. Although the ellipse is very nearly circular, the extreme distances 
of the earth from the sun differ by quite a considerable quantity, namely 
three million miles. 

KEPLER'S LAWS 

Tycho Brahe had refused to accept the Copernican view. Rejecting the 
Ptolemaic epicycles which gave no account of the relation of the behaviour 
of the planets to their propinquity to the sun, he adopted a compromise 
essentially the same as the earlier Egyptian doctrine. The earth remained 
at the centre, but the planets revolved around the sun. The recognition 
that the sun's apparent annual motion is not a perfect circle was a severe blow 
to idealistic dogmas which had reigned unchallenged for centuries. Ten 
years after the publication of his work on Mars, Kepler broke boldly away 
from the geocentric view of the universe and restated the doctrine of Aris- 
tarchus and Copernicus in the three laws which usually bear his name. 
These are : 

1. The earth and the planets move in ellipses with the sun at one focus. 

2. The radius vector of a planet's motion describes equal areas in equal 

times. 

3. The square of the sidereal period (T) of any planet bears a constant 

ratio to the cube of its mean distance (d) from the sun, i.e. 



If the year is the unit of time and the mean distance of the earth from the 
sun is taken as the unit of distance, the periodic times and mean distances 
of the planets at Kepler's time are given approximately in the following 



216 Science for the Citizen 

table, from which you will see that their ratio is constant to about the same 
degree of accuracy as the figures themselves. 

Planet T d T*+d* 

Mercury 0-24 0-39 0-971 

Venus 0-61 0-72 0-997 

Earth 1-00 1-00 1-000 

Mars 1-88 1-52 1-006 

Jupiter 11-86 5-20 1-000 

Saturn 29-46 9-54 1-000 

Kepler's account of the movements of the planets was not more satis- 
factory than the hypothesis of Ptolemy because the case was stated with 
greater logical subtlety, because it placed greater reliance on mathematical 
ingenuity, or even because it was simpler to grasp. It was more satisfactory 
because it could help people to do things. If we want to know when and where 
to put a telescope to see a planet in the sky, Kepler's recipe helps us in situa- 
tions when Ptolemy's recipe would lead us astray. If we want to know when 
and where to look for an occultation of Mars, Kepler's hypothesis shows 
us how to make an almanac which will be serviceable for a long while ahead. 
Ptolemy's hypothesis could not. A scientific hypothesis is not a passive 
prediction of future events. It is an active prescription for human conduct. 
For a long while to come the Copernican doctrine in its new form was still 
open to the objection that no annual parallax of the fixed stars could be 
detected by the instruments then available. Its universal acceptance by the 
scientific world was ensured by the fact that the earth's diurnal rotation and 
the laws of Kepler received independent confirmation from new knowledge 
about laws of motion to be explained in the next chapter. The recognition 
that the apparent paths of the sun and planets are not perfect circles was a 
far more drastic step than we might suppose. In the age of Kepler, science 
was only beginning to shake off the Platonic teaching which seeks to arrive 
at truth by logical argument based on self-evident principles and verbal 
definitions. Plato taught that the heavenly bodies must move in circles 
because the circle is the most perfect figure. Ptolemy had founded his system 
on the self-evident principle that the whole celestial sphere rotates around 
the earth. Arguing from these premisses his logic was flawless. The success 
of Kepler's calculations started a steady decline in the belief that logic is a 
sufficient guide to truth. 

To recapture the atmosphere of the time we may recall a type specimen 
of Aristotle's astronomy: " The shape of the heavens is of neccessity 
spherical, for that is the shape most appropriate to its substance and also by 
nature primary. . . . Every plane figure must be either rectilinear or curvi- 
linear. Now the rectilinear is bounded by more than one line, the curvilinear 
by one only. . . . If then the complete is prior to the incomplete, it follows on 
this ground also that the circle is primary among figures, and the sphere holds 
the same position among solids. . . . Now the first figure belongs to the first 
body, and the first body is that of the farthest circumference." Contrast these 
words with those of Francis Bacon: "It cannot be that axioms discovered 
by argumentations should avail for the discovery of new works; since the 



Children of the Sun 217 

subtlety of nature is greater many times over than the subtlety of 
arguments. . . . Radical errors in the first concoction of the mind are not 
to be cured by the excellence of functions and remedies subsequent. . . . 
We must lead men to the particulars themselves, while men on their side 
must force themselves for a while to lay their notions by and begin to 
familiarize themselves with the facts." 

The supremacy of fact over logic, the use of logic as an instrument to be 
calibrated continually by recourse to fact, is a lesson which every branch of 
science has had to master. Standing on the threshold of our own culture Roger 
Bacon declared: "If I had my way, I should burn all the books of Aristotle, 
for the study of them can only lead to a loss of time, produce error, and 
increase ignorance." One by one the natural sciences have abandoned the 
self-evident principles of Aristotle. Logical deductions from the self-evident 
truth that nature abhors a vacuum have proved unable to suggest the proper 
way of pumping water out of a mine, and the self-evident truth that bodies 
fall where they belong is not particularly useful when we come to calculate 
the path of a projectile. The past history of human culture shows us that a 
high level of scientific attainment in isolated fields has flourished when men 
have been forced to bring their beliefs to the bar of fact and use them as 
recipes of social conduct. That the social sciences have not passed out of 
the stage from which the natural sciences were emerging in the time of 
Kepler is sufficiently evident from the following exposition of The Nature 
and Significance of Economic Science by Professor Robbins : 

We have not yet discussed the nature and derivation of Economic Laws. . . . 
It will be convenient, therefore, at the outset of our investigations, if, instead 
of attempting to derive the nature of Economic Generalizations from the pure 
categories of our subject-matter, we commence by examining a typical speci- 
men. . . . It is a well-known generalization of elementary Price Theory that, 
in a free market, intervention by some outside body to fix a price below the 
market price will lead to an excess of demand over supply. . . . Upon what 
foundations does it rest? 

It should not be necessary to spend much time showing that it cannot rest 
upon any appeal to History. The frequent concomitance of certain phenomena 
in time may suggest a problem to be solved. It cannot by itself be taken to 
imply a definite causal relationship. ... It is one of the great merits of the 
modern Philosophy of History that it has repudiated all claims of this sort, and 
indeed makes it the fundamcntum divisionis between History and Natural 
Science that history does not proceed by way of generalizing abstraction. . . . 
It is equally clear that our belief does not rest upon the results of controlled 
experiment . . . our belief in this particular generalization and many others 
is more complete than belief based upon any number of controlled experi- 
ments. . . . But on what, then, does it depend? ... In the last analysis, 
therefore, our proposition rests upon deductions which are implicit in our 
initial definition of the subject-matter of Economic Science as a whole. 

This passage briefly summarizes every attitude to knowledge discarded 
by the natural sciences in reaching the prestige they now enjoy. The natural 
sciences owe their prestige to the fact that they provide man with the means 
of regulating his social conduct. They are able to do this because science 
rests on a wholesome distrust of logic, except in so far as the results of a 



2i8 Science for the Citizen 

logical process are tested continually by return to the real world. Sprat, 
Bishop of Rochester, in the first history of the Royal Society., which he 
speaks of as an "enterprise for the Benefit of Human Life by the Advance- 
ment of Real Knowledge" devoted the beginning of his narrative to the way 
in which real knowledge is gained. His resume of the scholastic tradition is 
worth quoting side by side with the preceding extracts from a contemporary 
economist. 

They began with some general definitions of the Things themselves, accord- 
ing to their universal Natures, then divided them into their Parts, and then 
out into several propositions, which they laid down as Problems: These they 
controverted on both sides and by many niceties of Arguments, and Citations 
of Authorities, confuted their Adversaries and strengthened their own Dictates. 
But though this notional War had been carry'd on with far more care and 
calmness amongst them than it was. Yet it was never able to do any great 
Good towards the Enlargement of Knowledge, because it rely'd on general 
Terms which had not much Foundation in Nature and also because they took 
no other Course but that of Disputing. 

Some folk seem to think that the human imagination is impoverished by 
accepting the supremacy of fact over mere logic. So it is healthy to recall 
the eloquent fervour with which Kepler began the announcement of his 
doctrine : 

What I prophesied two-and-twenty years ago, as soon as I discovered the 
five solids among the heavenly orbits what I firmly believed long before 
I had seen Ptolemy's Harmonies what I had promised my friends in the 
title of this book, which I named before I was sure of my discovery what 
sixteen years ago I urged as a thing to be sought that for which I joined 
Tycho Brahe, for which I settled in Prague, for which I have devoted the best 
part of my life to astronomical contemplations, at length I have brought to 
light, and recognized its truth beyond my most sanguine expectations. It is not 
eighteen months since I got the first glimpse of light, three months since the 
dawn, very few days since the unveiled sun, most admirable to gaze upon, burst 
upon me. Nothing holds me; I will indulge my sacred fury; I will triumph 
over mankind by the honest confession that I have stolen the golden vases of the 
Egyptians to build up a tabernacle for my God far away from the confines of 
Egypt. If you forgive me, I rejoice; if you are angry, I can bear it; the die is 
cast, the book is written, to be read either now or by posterity, I care not 
which; it may well wait a century for a reader, as God has waited six thousand 
years for an observer ! 



PRECESSION AND THE DATE OF ANCIENT MONUMENTS 

In Chapter I reference has been made to the work of Sir Norman Lockyer 
and others on early calendrical monuments. With the aid of the methods 
dealt with in this chapter the principles of orientation can now be made more 
explicit. The possibility of dating archaeological remains of this type is based 
on alignment to greet the rising or setting or in the case of the Great 
Pyramid transit of stars. The azimuths and transit elevation of stars depend 
on their declination which changes in the course of centuries owing to pre- 



Children of the Sun 



219 



cession. While the positions of the stars with reference to the plane of the 
ecliptic do not change, the axis of the celestial equator rotates about the 
ecliptic axis (Fig. 138) at an angle (the obliquity (e) of the ecliptic) which 
varies only very slightly in the course of millennia between the limits 22 35' 
and 24 13' during the whole period of historic time. According to the 



Poll Star now 



4 ^ /rude. ^-- V f ^ / 




FIG. 138. THE PRECESSION OF THE EQUINOXES 

The position of the stars with reference to the plane of the celestial equator slowly 
changes j but remains constant with reference to the plane of the ecliptic. From the 
standpoint of an earth-observer (i.e. according to the geocentric view) this slow change 
is equivalent to a rotation of the plane of the celestial equator about the pole of the 
ecliptic at approximately constant inclination (23i) to the ecliptic. Consequently the 
celestial pole revolves around the pole of the ecliptic at the same angle., and any bright 
star on the circle of its revolution will be a Pole Star at some time during the course of 
a complete cycle of about 26 3 000 years. The point of intersection of the equator and 
ecliptic shifts along the ecliptic in the same direction as the diurnal motion of the 
celestial sphere. Hence the vernal equinox which in the time of Hipparchus occurred 
when the sun occupied the same relative position as Aries now occurs when the sun 
is in Pisces, though the node<y is still called "The First Point of Aries." The two 
"Pole Stars" shown are, of course, different ones not two positions of the same one. 

Ptolemaic view (Fig. 138), the nodes or points of intersection of the sun's 
apparent path with the celestial equator shift in the course of a year about 
50 seconds of an arc along the ecliptic circle in the direction opposite to the 
sun's annual retreat. According to the Copernican view (Fig. 139), the pre- 
cession of the equinoxes results from the slow rotation of the earth's polar 
axis about the axis of the ecliptic at approximately constant inclination (e) 



220 



Science for the Citizen 



to the latter. The result is that the sun's position among the fixed stars on the 
vernal equinox has shifted from the constellation of Aries to Pisces since 
the time of Hipparchus, though the node is still called "the first point of 
Aries" denoted by the zodiacal sign for the ram (T). 

The explanation of the geocentric view of planetary motion given in this 
chapter is based on the observed change in the R.A. of a planet. The trace 
obtained is not the true orbit, but the projection of the orbit on the plane of 
the celestial equator. Actually the planets move close to the plane of the 
ecliptic, and a closer approximation to the true orbit is obtained by plotting 
their movements in celestial longitude. As stated on p. 105, celestial longi- 




365^ &y $ 



March 21, 
Sun over Equator 



June 2.1, 

Sun over 
T. of Cancer 



7o Constdhdmri- of 



FIG. 189. THE PRECESSION OF THE EQUINOXES 

From the standpoint of a sun observer (heliocentric or Copernican view) the Pre- 
cession can be regarded as analogous to the wobble of a spinning top. The earth's 
polar axis rotates slowly about the pole of the ecliptic and the earth's equator revolves 
at the approximately fixed inclination of 23J to the plane of the earth's orbit. This 
rotation means that the north pole, which is tilted sunward when the sun is "in Cancer," 
will be tilted away from the sun in Cancer a half-cycle (j. 2 6000 = 13,000 years) later. 

tude and latitude have the same relation to the axis and plane of the ecliptic 
as R.A. and declination to the polar axis and the celestial equator. Great 
circles of longitude intersect at the pole of the ecliptic just as great circles 
of R.A. (Fig. 141) intersect at the celestial pole., and longitude is measured 
like R.A. eastwards from p . Latitude is measured by elevation above the 
plane of an ecliptic just as declination is measured by elevation from the 
celestial equator, and colatitude (90 Lat.) is analogous to polar distance. 
Though this system is less useful for ready calculations for geographical 
use, it has special advantages for observations carried on over long periods, 
because the position of the stars with reference to the plane of the ecliptic is 
fixed. So the latitude of a star, i.e. its elevation above the ecliptic plane, is 
not affected by precession and its longitude increases 50J seconds of an arc 



Children of the Sun 



221 



per year. The sun's celestial latitude is always zero, and its motion in longi- 
tude is more nearly uniform than its motion in R.A. If we know the latitude 
and longitude of a star at one date we can therefore find it easily at another, 
and we can calculate the change in R.A. and declination due to the rotation 
of the nodes if we know how to convert observations from one system of co- 
ordinates to the other. We can then calculate the azimuth and times of rising 

Sim. ;Marck 21 

Dedw.O" Lut.O* 
R.A.Oh. 



o 



&' 



Sun Dec. 21 / / 

KAjsh^ioL^yo 

AiIi/> < 



AbcfN ..4"^-,\ 

/ < ojv - I , -*"^ ~- v^7 . 

I .<H ' , X' 





Sun Juttc 21 

l \ Dedin 23k*M Latff 

\ K.A.6h. Lonq.90" 

> \ \\^t-> 'i 

I v~ -N 



ioyzg: ^/0- 

\ 
\ 

\ Pole 

\(RA.18h. 
"1.66 



Sim Sept. 23 

DedLn,.0 L&k.O" 
RA.l2.1i. Long. 130 

FIG. 140. ECLIPTIC AND EQUATORIAL CO-ORDINATES OF THE HEAVENS SUPERIMPOSED 

IN A PLANE 

By representing both the latitude and declination parallels as circles in conformity 
with Fig. 51 a distortion of the great circles of longitude and Right Ascension is 
produced. 

or setting of stars at different periods in the history of the earth. The solution 
of the problem also illustrates the method for determining the true orbit of 
a planet. 

Since longitude and R.A. are both reckoned from the intersection (T) of 
the plane of the ecliptic and equator, i.e. the sun's position on the vernal 
equinox, T is 90 from the pole (E) of the ecliptic along the great circle of 
longitude and 90 from the celestial pole (P) along the great circle R.A. Oh 



222 Science for the Citizen 

(see Fig. 141). The pole of the ecliptic, the celestial pole, and T 5 forni three 
corners of a spherical triangle of which 

EP is the colatitude of the celestial pole or the polar distance of the 
ecliptic pole and is equal to the obliquity of the ecliptic (approxi- 
mately 23'). 

ET 90 = PT 

/ T EP ^ the i on gitude of P 
/ TPE = 360 - R.A. of E. 

So applying the cosine formula for spherical triangles : 

cos ET = cos EP cos PT + sin EP sin PT cos TPE 
/. cos 90 = cos 23i cos 90 -f sin 231 sin 90 cos TPE 
/. cos vPE = 

/. TPE = 90 

Similarly 

cos PT = cos ET cos EP + sin ET sin EP cos TEP 
.*. cos 90 = cos 90 cos 23 + sin 90 sin 23J cos TEP 
/. cos TEP /. TEP = 90" 

Thus the triangle EPT in Fig. 141 is a right angled triangle in which the 
side EP is (approximately) 23-|, the sides ET and PT are each 90 and the 
angles E and P are also each 90. A spherical triangle is also formed with the 
three corners E, P, S, corresponding to the poles of the ecliptic and of the 
celestial equator and a star S. The two sides ES, PS, are arcs along the longi- 
tude and R.A. meridians of the star respectively. Since TEP = 90 and TES 
is the longitude of the star, 

PES TEP TES 
= 90 Long, of S 

Since TPE = 90 and TPS is the R.A. of the star, 

EPS =- TPE f TPS 
- 90 | R.A. of S. 

In the spherical triangle EPS therefore we have five quantities of which any 
two may be found, if the other three are known. They are : 

EP 23i approximately 

ES = 90 Lat. (= colatitude) 

PS = 90 declin. (-polar distance) 

PES - 90 - Long. 

EPS -90 I R.A. 

Applying the cosine formulae : 

(i) cos PS cos EP cos ES -(- sin EP sin ES cos PES 

/. sin (declin.) = cos 23i sin (lat.) + sin 23| cos (lat.) sin (long.) 

(ii) cos ES = cos EP cos PS + sin EP sin PS cos EPS 

/. sin (lat.)=cos 23-| sin (declin.) sin 23i cos (declin.) sin (R.A.)* 

* Since cos (90 + A) -- - sin A. 



Children of the Sun 



223 




Suns' position 

March Zl. 

FIG. 141. THE Two SYSTEMS OF CELESTIAL CO-ORDINATES 

Celestial latitude and longitude give the direction of a star as seen from the earth's 
centre with reference to the plane of the ecliptic (or earth's orbit about the sun). 
Declination and R.A. give its direction as seen from the earth centre with reference 
to the plane of the celestial equator (at right angles to the earth's polar axis). Since 
the position of the stars with reference to the ecliptic plane is fixed the latitude of a 
star is not affected by precession. Longitude is reckoned like R.A. from the meridian 
which passes through the node nr . Since this slowly retreats at 50J seconds of an arc 
per year the longitude of any star increases by 50 J" per year. Compare this figure 
carefully with Fig. 48 in Chapter 2 by tilting either through U3J. 

So if we know the R.A. and declination of a star we can calculate its celestial 
latitude, and if we know the celestial longitude and latitude of a star we can 
get its declination by the cosine formula. Similarly, applying the sine formula : 

sin EPS sin PES 



sin ES 

sin (1)0 + R.A.) 


sin PS 
sin (90 long.) 


sin (90 - lat.) 
cos (R.A.) 


sin (90 declin.) 
cos (long.) 


cos (lat.) ~~ 

fw\ e ("& A "\ 


cos (declin.) 
cos (long.) cos (lat.) 



and 



(iv) cos (long.) = 



cos (R.A.) cos (declin.) 
cos (lat.) 



224 Science for the Citizen 

For a check to the formula we may calculate the azimuth of the rising May 
year (p. 69) sun at the terrestrial latitude (51 approximately) of Stonehenge. 
To do this we need to know the sun's declination on May 6th. The May year 
begins at a date half-way between the vernal equinox and summer solstice. 
So the longitude is 45. The sun's celestial latitude is always zero. Thus 

sin (declin.) = cos 23f sin + sin 23i cos sin 45 
= + 0-3987 X 0-7071 " 
= + 0-2819 
declin. - + 16J 

At 51 (terrestrial) latitude the azimuth of rising is given by the formula: 

sin (declin.) 



cos (azim.) = 



cos (lat.) 
sin 16 



cos 51 

-=0-448 

/. azimuth = 63| (approximately), or 26i north of the 
east point. Allowing for a slight change in the obliquity of the ecliptic, this 
corresponds to the older May year alignment which Lockyer describes in 
addition to the solstitial setting (Fig. 1). 

As a problem in assigning a date to archaeological remains the Great 
Pyramid will serve for illustration. This depends on the inclination of the 
tunnel to greet the transit of Soth> or Sirius as we now call it. To simplify 
the issue we may calculate the declination of Sirius 4,100 years ago, i.e. in 
2164 B.C. The present declination of Sirius is (approximately) 16| South, 
and its R.A. is 6 hours 42 minutes (approximately), or 101. To get its 
co-ordinates at the date mentioned we may first refer its position to the 
plane of the ecliptic, thus : 

(i) sin (lat.) - cos 23| sin (- 16|) - sin 23 cos (~16J) sin 101 

= cos 23-| sin 161 sin 23 cos 16-J cos 11.* 
= - 0-6358 
/. lat. -= 39-5 

cos(-~ 16-1) cos 101 
(ii) cos (long.)-: -I -- v 2 } 



cos (39-5) 

_ cos 16i sin 11 
_ _____ 

== 0-2371 

Since cos 76-3 = 0-2371 (approx.) 
cos (180 76-3) 0-2371 
.'. longitude 103-7 

The precessional shift is 50 seconds of an arc in the longitude of all stars 
per year. Hence the increase since 2164 B.C. is (50J X 4100)" or 

* Since sin (90 + A) = cos A. 



Children of the Sun 



225 



50 x 4100 -T- 3600 degrees, i.e. 57-2. Thus the longitude of Sirius 
at that date was 103-7 -~ 57-2 =- 46-5. So that the declination of Sirius 
at that time is given by : 

sin (declin.) = cos 23J sin (39-5) +sin 23| cos (39-5) sin (46-5) 
= cos 23 1 sin 39-5 + sin 23 J cos 39-5 sin 46-5 
^0-5833+0-2232 
= 0-3601 
/ . declination =21-1 S. 

The accompanying figure (Fig. 142), which should be compared with Fig. 2, 
shows that if the angle of the Pyramid is approximately 52, the beam of 



\*$ 

%v '% 

V Ik V 




FIG. 142. THE GEOMETRICAL RELATIONS OF THE TRANSIT OF SIRIUS TO 
THE GREAT PYRAMID 



Sirius striking its face at right angles when crossing the meridian is elevated 
38 from the horizon. The terrestrial latitude of the Pyramid is 30, so that 
the angle which the star made with the celestial equator (i.e. its declination 
at the time when it was built) was approximately 22% which differs by less 
than a degree from the figure calculated, for the date assigned, with the aid 
of four-figure tables and approximate values for the obliquity of the ecliptic 
and the angle of the Pyramid. Proceeding in this way we get 20-8 in 
2064 and 23 6 in 2864 B.C. for the declination of Sirius. If, then, the tunnel 
was in line with the transit of Sirius, the Pyramid must have been com- 
pleted about 2400 B.C. To assign a more precise date a graph of more accurate 
values of declinations at different dates, as given in Lockyer's book on 
Stonehenge y can be prepared, and the date when the declination of the star 
agreed most closely with the architectural data can be read off from it. 

The possibility of dating a solar monument like Stonehenge depends on a 
different principle. What we call the summer solstice is the day on which the 
positive declination of the sun is equivalent to the angle of obliquity, and since 
the inclination of the equator to the ecliptic is not involved in the precession 

H 



226 Science for the Citizen 

of the equinoxes, the precessional rotation does not affect the sun's declination 
and therefore its azimuth of rising and setting on the solstice. Actually, the 
observations which have accumulated from the time of the Alexandrian 
Hipparchus and his Babylonian predecessors show that the obliquity of the 
ecliptic has changed very slightly less than a degree since the beginning 
of recorded history. Hence the sun's declination, which is now about 23| 
on the solstice, was about 24 at the time when Stonehenge was erected, and 
this makes a perceptible azimuth difference which can be detected by the 
alignment of an avenue placed to meet the rising midsummer sun. 



EXAMPLES ON CHAPTER IV 

For these the reader will need Whitaker's Almanack and a planisphere, which 
can be obtained from an educational bookseller for about 2s. 6d. or half a 
dollar. 

1 . With the aid of a map on which ocean distances are given, work out 
the distances by great-circle sailing from ports connected by direct routes. 
What are the distances between London and New York, London and Moscow, 
London and Liverpool? 

2. On April 26th, the sun's R.A. being 2 hours 13 minutes, the bearings 
of three stars were found to be as follows, by a home-made instrument : 

Azimuth Zenith Distance Local Time 

Pollux W. 80 from S. 45 9.28 p.m. 

Regulus W. 28 from S. 41 9.39p.m. 

Arcturus W. 7 from N. 31 12.50p.m. 

Find the decimation and R.A. of each star and compare these rough estimates 
made at Lat. 50 J N. with the accurate determinations given in Whitaker's 
Almanack. 

3. To get the exact position of the meridian a line was drawn between two 
posts in line with the setting sun on July 4th at a place Lat. 43 N. At what 
angle to this line did the meridian lie? (Whitakcr gives the sun's declination 
on July 4th as 23 N.) What was the approximate time of sunset? 

4. If the R.A. of Sirius is 6 hours 42 minutes and its declination is 16 S, 
find its local times of rising and setting on January 1st at 

Gizeh Lat. 30 N. 

New York Lat. 41 N. 
London Lat. 51 -J' J N. 

Check with planisphere. 

5. Find the times of rising and setting of the sun on February 9th 3 also the 
azimuth of rising and setting on the same date at the latitudes of London, New 
York, and Cape Town. 

6. Find the times of rising and setting of Vega, a week before and a week 
after the vernal equinox at Lat. 51 N. Compare them with the times of sunrise 
or sunset. Give the azimuths of rising and setting at the same latitude. (R.A. of 
Vega 18 hours 35 minutes, Declination 38 43 '.) 



Children of the Sun 227 

7. From the tables of R.A. and Declination in Whitaker, make a graph 
of the times of rising and setting of any star and of the sun throughout the 
year at your own latitude. Hence find the date on which the star is seen rising 
and setting in morning twilight (one hour before sunrise) or evening twilight 
(one hour after sunset). Check with a planisphere. 

8. Find the declination of Polaris and y Draconis in the time of Hipparchus 
(150 B.C.). 

9. According to Fotheringham, the Chaldean astronomer Cidenas taught 
that Mercury was never more than 22 distant from the sun. If its orbit were 
circular what would be the relative distances of Mercury and the earth from 
the sun? 

10. With the aid of three consecutive issues of Whitaker tabulate the R.A. 
of Venus and Mars for a sufficient period to determine the sidereal period of 
each planet, and determine the relative breadth of their heliocentric orbits, 
assuming that they are the circular form. 

11. Use the sidereal periods of the moon and one of the planets Venus 3 
Mars or Jupiter,, the R.A. of the moon and the R.A. of the planet chosen as 
given for the first day of the month in Whitaker,, to calculate approximate 
times of lunar conjunction and opposition during the ensuing lunar cycle. 
Check from the same source. 



THINGS TO MEMORIZE 

1. Solution of spherical triangles. 

cos a cos b cos c + sin b sin c cos A 
sin a sin C 



sin A= 



2. If Azim. is reckoned from the south point: 

sin Decl. sin Lat. cos Z.D. cos Lat. sin Z.D. cos Azim. 

When a heavenly body is rising or setting 

sin Decl. = cos Lat. cos Azim. 

. TT A sin Azim. sin Z.D. 

:*. sin Hour Angle =- 

cos Decl. 

4. If Azimuth is reckoned from the south point, cos Azim. is positive for 
stars with southerly, and negative for stars of northerly, bearing. Sin Azim. is 
reckoned negative when a star is east of the meridian. 

_ . . _ .,. distance between foci 

5. Eccentricity of ellipse : : 

major axis 



/minor axis\- 



\ major axis/ 

6. Kepler's Third Law. If T is the sidereal period of a planet and d its 
distance from the sun, T 2 -r d 3 is the same for all the planets. 

7. See also the four formulae on pp. 222-3 relating celestial latitude and 
longitude to declination and R.A. 



CHAPTER V 

WHEEL, WEIGHT, AND WATCHSPRING 

The Laws of Motion 

THE determination of longitude in modern navigation and exploration depends 
upon comparing local time as determined by the sun or stars with time at 
Greenwich, as explained on p. 79. This method has only been in use a 
hundred and fifty years. Before then, it was not possible to construct clocks 
with sufficient reliability to keep in step with standard time during a sea 



hour 
candle 





FIG. 143. PRIMITIVE TIMEPIECES 

Before the introduction of clocks the early monasteries of northern Europe regulated 
the routine of daily ritual by the use of hour candles. The word clock is derived from 
cloche, a bell. The bell which tolled matins and vespers was once the village clock. 
Hour glasses of great antiquity were comparable to the device still used to time the 
boiling of an egg. 

voyage. The laborious method of lunar distances or the use of eclipses and 
occultations, when they happened to be visible, were the only methods 
available when long-distance westerly courses were first undertaken in the 
closing years of the fifteenth century. For practical purposes the use of the 
moon as a celestial signpost had two immense disadvantages. One is that 
clear-cut moon signals suitable for observation with rough-and-ready in- 
struments at sea are comparatively rare events. The other is that methods of 
finding longitude by relying on the moon's position (as illustrated in Fig. 37 
and Fig. 110) are liable to involve a large error unless correction is made for 
parallax. Thus the method of Amerigo cited on p. 180 gives a very poor 
result, and its use in the Spanish and Portuguese fleets would have been 



Wheel, Weight, and Watchspring 229 



-n +. i x- Circular 

tetdv* jyial 

to carkf/oaf 




Cork 

Float ^R 



design 



FIG. 144. WATER CLOCKS 

The clepsydras of the early Mediterranean civilizations were simply graduated vessels 
of conical shape to ensure a steady rate of flow as they emptied. They were filled 
daily. The mechanics of Alexandria made more elaborate devices which, so far as we 
can be sure from the descriptions available, seem to have employed two principles. 
One was the float with an upright rod. This might bear a simple pointer which recorded 
the hours on a vertical scale. It might have a ratchet which worked a cog wheel carrying 
a pointer for a circular dial. Alternatively a revolving mechanism something like a 
water wheel in design was used. Water clocks of this type were still used in seventeenth- 
century Europe. The float clock shown in this figure is provided with a reservoir 
designed to maintain a fixed rate of flow. Hence it could be used to record hours of 
equal length. Alexandrian clocks were not so made. Before the invention of the Arab 
sundial with its style pointing to the celestial pole the interval from sunrise to sunset 
was taken as a unit of time, and the hours were not equivalent in different seasons. 
In ancient clepsydras, an adjustment was made by a plunger which varied the capacity 
of the vessel, and hence the rate of flow from it. The bent tube on the right side of 
the float clock acts as a siphon which automatically empties the vessel, and thus makes 
possible the continuous record of time without personal attention. 



230 



Science for the Citizen 



impossible without the assistance of trained astronomers who accompanied 
their expeditions. 

In 1714 the British Government offered a prize of 20.,000 for the inven- 
tion of any means of determining longitude at sea with an error of not more 
than 30 nautical miles. The prize was awarded for Harrison's chronometer 




FIG. 145. The water clock of Ctesibius invented about 250 B.C. had water dropping like 
tears into a funnel from the eyes of a statue. A float mechanism raised another human 
figure with a pointer which indicated the hours on a vertical cylinder. Once in twenty- 
four hours the figure descended to the bottom of the column by a siphon mechanism 
as shown in the previous illustration. The siphon outflow worked a water wheel 
which very slowly rotated the cylinder dial, making a complete rotation in a year. The 
graduation of the cylinder was adapted to the varying lengths of the hours throughout 
the seasons. The gearing is not shown. The curious feature of this device is the applica- 
tion of so much ingenuity to conserve the seasonal hours which preceded the invention 
of the Arab sundial. It is said to have been installed in a temple. 

in 1765. Simultaneously, Le Roy of Paris invented a better device which is in 
principle like the modern chronometer. These patents were the crowning 
achievement of sustained effort over a long period. As early as 1530,, Gemma 
Frisius, a Dutch mathematician and cosmographer, had pointed out the great 
advantage of a chronometric method (p. 79) for finding longitude at sea. 
Thereafter the improvement of the clock constantly engaged the attention of 



Wheel) Weight^ and Watchspring 231 

the leaders of physical science. Galileo, Hooke, and Huyghens, who laid the 
foundations of the dynamical principles systematized by Newton in the 
Principia, were actively concerned with the betterment of the timepiece; and 
some of the most fruitful theoretical work was a by-product of their interest 
in doing so. One of the last acts of Galileo, who discovered the principle of 
the pendulum, was to dictate a project for the construction of a pendulum 
clock. Hooke, who is said to have first introduced the hair spring, also in- 
vented the "anchor escapement" and a device for cutting watch wheels 
accurately. In a very real sense, his Law of the Spring was a by-product 
of the technology of time-keeping. Huyghens, who first developed the theory 
of centrifugal motion, published it in his Horologium Oscillatorium which 





FIG. 146 

The design of these two clocks illustrates vividly the dual social context in which 
the technology of mechanical timekeeping progressed in the middle ages. 

describes his invention of the pendulum clock. He also made several attempts 
to make a seaworthy clock suitable for finding longitude by the method of 
Frisius and subjected his models to tests at sea. 

The first clocks in the modern sense, i.e. clocks geared with toothed 
wheels, were driven by the falling of a weight, and checked daily by the noon 
sun. They were large clumsy affairs, scarcely as accurate as the magnificent 
but quite immovable water clock or clepsydra which was constructed by 
Ctesibius at Alexandria (see Fig. 145) in the third century B.C. Although 
designs were put forward at an earlier date, weight-driven clocks did not 
come into use before about A.D. 1000, when they were first put up in churches 
and monasteries. For two centuries their use was confined to the official 
timekeepers of Christendom. Till about A.D. 1450 they were not sold for 
secular use except for installation in public buildings. In England the first 
public clock was erected in A.D. 1288. Meanwhile Arabic astronomy was 



232 Science for the Citizen 

beginning to filter into northern Europe. In the generation which preceded 
Copernicus, Walther of Nuremburg (1430-1504) equipped what appears to 
have been the first observatory with mechanical clocks. Spring-driven 
portable watches the so-called "Nuremburg eggs" were marketed at the 
beginning of the sixteenth century. 

In previous chapters enough has been said to emphasize the importance 
of an accurate and portable device for registering time as a basis for astro- 
nomical measurements which underlie calendrical practice and navigation. 
Mechanical clocks made no mean contribution to the great progress of 
astronomy during the epoch of the great navigations. Aside from its usefulness 
as an instrument of observation, the wheel-driven clock created a new 
problem in mechanics. Finding the laws which describe the motion of the 
clock also disclosed new laws which describe the motions of the solar system. 
Though the telescope did much to encourage a favourable reception for the 
doctrine which Copernicus had revived, the invention of the pendulum clock 
was the incident which gave the death-blow to the geocentric principle. 

The Copernican doctrine, and still more the laws of Kepler, equipped 
navigational astronomy with more reliable and manageable rules for calcu- 
lating conjunctions of planets. Apart from the convenience of doing so, 
there is nothing to explain the popularity of the Copernican teaching. 
A century after the death of Copernicus no known facts of terrestrial 
experience could be brought forward to justify the major premises on 
which his doctrine rests. The supposed orbital motion of the earth was 
difficult to reconcile with failure to measure the annual parallax of a 
star, a feat which was not accomplished till about 1830, when improve- 
ments in optical instruments made it first possible. Telescopic observations 
like Galileo's descriptions of sunspots suggested the diurnal rotation of other 
heavenly bodies before it was possible to point out a single occurrence due 
to the earth's axial motion. Even when Kepler published his final treatise 
on planetary orbits, scepticism was common among his contemporaries; and 
Bacon, whose Novum Organum was written as a challenge to the Aristotelian 
tradition, obstinately rejected the heliocentric view. From a landsman's point 
of view, the earth remained at rest till it was discovered that pendulum clocks 
lose time if taken to a place nearer the equator (see p. 288). After the invention 
of Huyghens the earth's axial motion was a socially necessary foundation for 
the colonial export of pendulum clocks. 

The mechanical principles discovered by Newton's immediate prede- 
cessors, Stevinus, Galileo, Hooke, and Huyghens, bring us face to face with 
one of the great dichotomies in the history of human achievement. While 
science in the ancient world was chiefly concerned with heavenly motions, 
that of our own time is largely occupied with earthly ones. This neglect 
of terrestrial motion by men of science in antiquity is not surprising. Neither 
their technical equipment nor their social environment was propitious to 
studying it. Antiquity was ill supplied with mechanical devices for registering 
small intervals of time, and so long as civilized life was restricted to sunny 
climates there was no pressing need to invent them. While slave labour was 
abundant there was no social pressure to bother about how things get moved. 
Merchandise was carried by slave-driven galleys equipped with sails for use 



Wheel, Weighty and Watchspring 233 





FIG. U7 

(a) Nocturnal or Night Dial (about 1700). () A Nocturnal in Use. 

Reproduced by permission of H.M. Stationery Office from "Time Measurement," 
Part I. by Dr. F. A. B. Ward. 

H* 



234 



Science for the Citizen 




I 



c = 

H O 





I* 

; "o 



14 



Wheel, Weight, and Watchspring 235 

when winds were favourable. It was the business of the overseer to control 
the slave and of the priests to propitiate the winds. Science was equally 
irrelevant to both methods of persuasion. 

Northern civilization, where knowledge was rapidly advancing in the 
sixteenth and seventeenth centuries of the Christian era, presents a different 
picture. Mechanical timekeeping is a social necessity for a highly developed 
culture in the grey northern climate. During the decline of chattel slavery, 




FIG. 149. ASTRONOMICAL CLOCK OF THE SIXTEENTH CENTURY 

from the time of Constantine onwards, mechanism had penetrated everyday 
life in other ways. The water mill was replacing irksome toil. Gunpowder had 
entered the theatre of war. Marksmanship required a close-up view of motion, 
and metallurgical development (p. 363) received a new impetus, making new 
demands on water power. The circumstances of artillery practice, of clock 
manufacture, of navigation, and of water power, belong to a social context in 
which men of capacity could find wealthy patrons to encourage mechanical 
enquiries. The new power with which gunpowder equipped imperialist 
designs on the New World, the increasing importance of astronomy in 
westerly navigation, and the introduction of the clock into secular use, were the 



236 



Science for the Citizen 




FIG. 150. EARLY SEVENTEENTH-CENTURY OBSERVATORY WITH SIMPLE QUADRANT 
(NON-TELESCOPIC) AND MECHANICAL CLOCKS 

The introduction of clocks into astronomical practice during the latter end of the 
fifteenth century increased the facility of determining the time of transit, and hence 
R.A. of stars. With more accurate clocks at their disposal in the middle of the seven- 
teenth century astronomers undertook extensive revision of the star catalogues. 
Prominent among them was Hevel of Danzig. His observatory is here seen. The 
catalogue of Hevel formed the basis of more refined observations with telescopic sights 
in the Paris and Greenwich observatories during the latter end of the seventeenth 
century. 

(Note that the accuracy achieved by medieval astronomers who worked like Tycho 
Brahe without telescopic instruments was due to the large scale of their instruments. 
Thus Tycho Brahe used a quadrant 19 feet across. A circle of radius 19 feet has a 
circumference (360 degrees) of 19 x 2 x 3-14 feet, i.e. roughly 120 feet or 4 inches 
per degree. Since it is quite easy to distinguish half a millimetre (one-fiftieth of an 
inch), an accuracy of from one hundredth to one two-hundredth of a degree could be 
obtained, if the scale was accurately calibrated.) 

principal circumstances which conspired to create the adventurous hopeful- 
ness of seventeenth-century capitalism, when Bacon penned his eloquent 
plea, "the true and lawful goal of science is that human life be endowed with 
new powers and inventions." 

The military manuals published from 1500 onwards (see Fig. 151) bear 
eloquent testimony to a new impetus in the social background of Galileo's 



Wheel, Weight, and Watchspring 237 

"Two New Sciences." For the first time in the history of mankind precise 
measurement of rapid long-range motion became a social necessity. In its 
turn this demanded instruments for measuring time in short intervals. In 





FIG. 151. MILITARY MATHEMATICS 

These two prints taken from old books show one way in which solving the problem 
of motion had become a technical necessity in the period of Stevinus and Galileo. 
The upper one is from Bettino's Apiaria (Bologna, 1645). The lower from Zubler's 
work on geometric instruments (1607). 

the sixteenth and seventeenth centuries the improvement of clock design 
was a matter of as much importance as the improvement of motor cars today. 
Exploration and navigation demanded a seaworthy timepiece, while marks- 



238 Science for the Citizen 

manship called for a new unit of time reckoning. The problem of longitude in 
westerly courses and the problem of range-finding in mechanized warfare 
converged to a common focus, when the second became a unit of time measure- 
ment for the first time in history, and north-western Europe found it possible 
to dispense with a religion of saints' days. 

MECHANICS IN THE NEWTONIAN AGE 

Many people find the mechanics of the Newtonian age difficult to under- 
stand. To avoid discouragement it is therefore best to realize how some of 
the difficulties arise. The custom of speaking of mechanics as one of the 
exact sciences it is even called "applied mathematics" conceals the truth 
that all true science is as exact as it needs to be in virtue of the social circum- 
stances which force it to tackle new problems and as exact as it can be with 
the instruments at its disposal. In reality the textbook mechanics of Newton 
and Galileo is grossly inaccurate as a description of most mechanical devices 
which we ourselves commonly meet. The mechanics of the age in which 
Galileo and Newton lived was good enough for what were then called 
machines. It was brilliantly successful when applied to the problems of 
celestial motion. In a sense it was less the corner-stone of modern mechanics 
than an eloquent obituary on the science of antiquity. The world had not 
yet begun to measure power produced without the aid of man or beasts. 

The problem of Newton's age was to connect the new motive power of 
the cannon ball with the behaviour of devices like levers and pulleys which 
distribute the power of human beings or of animals. Machines like dynamos 
and gas engines which generate power from inanimate sources were not yet 
known. The water wheel or windmill took power from Nature at no 
cost to human effort. Newtonian science had no balance-sheet for heat 
and work. You will not understand the mechanical principles of Galileo 
and Newton if you expect them to apply to the machinery of rapid large- 
scale industrialization in the Soviet Union, and you will be less worried by 
the words in which they were stated if you remember something else. Views 
which are now commonplace were entirely novel in the age of the first 
pendulum clocks. Because they were novel and because they were still in 
violent opposition to traditional belief it was pardonable to state them in a 
cut-and-dried way, neglecting qualifications which seem imperative to us. 

It will help you to get the atmosphere of the time, and to see what the last 
sentence means, if we begin with a principle which is implicit in Galileo's 
treatment of the path of the cannon ball. According to the Aristotelian 
teaching, which was adapted by St. Paul to explain the mystery of the Resur- 
rection, everything in the universe had its place. If things got out of place 
they returned sooner or later to the place where they belonged earthly 
bodies to earth, celestial bodies (including the "spirits" of the retort) to 
heaven. Like other self-evident principles on which economists rely, this 
had very little relevance to the conduct of secular life, unless taken to signify 
that smoking is a celestial occupation. In particular it does not help you to 
decide where a bullet belongs. So long as Aristotle's physics enjoyed as much 
authority as the Bible, it was a novel and exciting discovery to find that things 
just go on moving till something stops them. It was just as novel as Darwin's 



Wheel, Weight, and Watchspring 239 

doctrine when most Englishmen still believed the Biblical account of creation. 
The words in which Newton's generation announced the discovery are no 
more surprising than the fact that JHuxley's views about man's nearest 
Simian ancestors are not universally held at the present time. 



RiaLzrhere 




rider frcnn xiglit Zizie. at 
' 



FIG. 152. HITTING A MOVING OBJECT 

If the bullet is fired along the line of sight, it will miss. The rider will have moved 
on beyond where he was along the line of sight at the instant when the bullet left 
the muzzle. To hit the mark the gun must be tilted away from the line of sight in the 
direction of the rider's motion. If the rider moves at a velocity v feet per second, he 
covers vt feet (here 3z>, since the time is 3 seconds) in r seconds. In the same time 
the bullet travelling with velocity V feet per second covers a distance of Vt (here 3V) 
feet along its own path. If bullet and rider reach the same point, as when the bullet hits 
its mark, their paths make two sides of a right-angled triangle with the sight line at 
the moment of firing, enclosing the angle a which measures the tilt of the gun to the 
direct line of sight, when the aim is correct. The rider's path (vt) and the bullet's 
path (Vr) are respectively the perpendicular and hypotenuse to a, i.e. sin a vt ~- Vt or 

sin a = v ~ V = cos (90 a) 

The proper angle depends only on the speeds of the bullet and rider, and it is not 
necessary to know the distances. 

How the principle of inertia emerges from the social context of the time 
is shown in Figs. 152 and 153, which illustrate two of the most elementary 
problems of marksmanship. If you are standing still (Fig. 152) you have to 
shoot ahead of it to hit an object moving at right angles to your direct line of 



240 



Science for the Citizen 



vision at the instant when the charge explodes. Conversely, if you are moving 
at right angles to the direction in which you are sighting a still object, you 
have to tilt the gun to the rear (Fig. 153). If you aim straight at the object 
the bullet will be ahead of it by the time it is level with its mark. The bullet 




BiitZei affcr 2 sees. 



c >-- + Bulld after 1 sec. \ 

>k x 



readies 

here, wkerilnillet 
mark 

FIG. 153. AIMING AT A STILL OBJECT WHILE MOVING 

If the bullet is fired along the line of sight, it will miss the mark, because it moves as if 
in each second : (i) it continued to progress with the velocity v of the rider along its 
course (i.e. the motion it had while it was still in the muzzle), (H) it had also the same 
velocity, V, along the direction in which the gun points, as it would have if the rider 
were at rest. To hit the mark it must be tilted away from the direct line of sight in 
the direction opposite to the rider's motion. Travelling at V feet per second in line 
with the axis of the gun, it moves simultaneously v feet per second parallel to the 
rider. If the angle of tilt (a) from the sight line is correct, its velocity v along the 
rider's course must keep it in the direct line of sight at the moment of firing. Hence, 
if a is the right angle of aim, the distance (vt^ the rider would travel in time r, the 
distance (Vr) the bullet would travel if the marksman were at rest and the direct line of 
sight, together form a right-angled triangle, in which sin a = vt -f- Vr, or 

sin a = v -f- V = cos (90 a) 

Thus the same rule describes the correct angle of tilt whether the marksman is moving 
at a particular velocity or the object at which he aims is moving at the same velocity, 
one or the other being still. In other words the tilt of the gun is only concerned with 
their relative velocities. 

therefore behaves as if it kept its motion sideways at the moment of leaving 
the muzzle and gained at the same time a new motion along the line of aim, 
just as a boat drifting in a current (Fig. 154) keeps the motion it gets from 
the oars and gains a simultaneous drift down stream. 



Wheel, Weight, and Watchspring 241 

The fact that it does so was quite unexpected when the science of marks- 
manship was in its infancy. This you can judge from the opening sentence of 
the second chapter in the Leviathan: 

That when a thing is still, unless something else stir it, it will lie forever still, 
is a truth that no man doubts of. But that when a thing is in motion, it will 
eternally be in motion unless something else stay it, though the reason be the 
same, that nothing can change of itself, is not so easily assented to. 



Cork drifts 1% miles m 1 hr. 






FIG. 15-J. COMPOSITE MOTION OF BOAT IN CURRENT 



As the moving marksman must tilt his gun from the sight line, so an oarsman must 
keep the keel turned away from the direction in which the boat must progress to 
reach its destination in a straight line. The same construction for finding the correct 
angle fits either case. If the boat is to be rowed along a course at right angles to the 
current, the keel must be inclined at a to this correct course, and at (90 a) to the 
current. To make a scale diagram, lay off equivalent distances traversed by a cork 
drifting in the current and by the boat, rowing at its ordinary straight line velocity 
in still water. These must be the sides of a right angled triangle, or of a parallelogram 
whose diagonal is at right angles to the current. The figure is drawn for a three-hour 
course, during which the boat would row 7 miles in still water, and a cork would 
drift 4 miles. Only one right-angled triangle can be made with these two sides, if 
one is the hypotenuse. In this case, sin a = 4 4- 7J 0-6, which is the same as the 
ratio of the current velocity (#), and the velocity (V) of the boat in still water. In 
general, sin a = v -~ V = cos (90- a), so that a can be got from tables of sines. 
The resultant velocity is Vv 2 v*. 

These words, written by the materialist philosopher Hobbes when Newton 
was still a boy, are almost identical with those in which Newton himself 
stated his First Law of Motion. As they stand they are somewhat metaphysical. 
We have no experience of bodies moving with absolutely constant speed or 
of things being permanently at rest or of any process going on for ever. Still, 



242 Science for the Citizen 

the essential fact of the sluggishness of matter or inertia is a very tangible one, 
and directly contrary to Aristotle's teaching. We all know that when a train 
or bus stops or starts our bodies continue to move in the direction of the 
previous movement, or remain at rest, as the case may be. Our feet, firmly 
set on the floor, follow the movement of the vehicle. Consequently we lurch 
backwards when the train or bus starts, or forwards when it stops. In this 
instance the movement of the vehicle is usually straight at the time. In 
turning a corner we lurch forwards and so outwards, a difficulty which every- 
one has to learn to forestall when first attempting to ride a bicycle. Our 
tendency, which we soon learn to counteract by leaning inwards, is to go over 
on the outer side of the curved path traversed by the machine. This illustrates 
the English idiom which embodies the most important aspect of inertia. We 
"fly off at a tangent." The principle of inertia signifies that when we turn a 
bend, we have to exert a pull by bending inwards to prevent ourselves con- 
tinuing to move in a straight line forwards. 

You may properly object that the flywheel of a steam engine does not go 
on for ever, and that Newton's first law is therefore plain nonsense. This 
would be wrenching it out of the context to which it belongs. Newton's 
generation knew as well as we do that an iron ball does not roll along a 
stretch of ice indefinitely. They also noticed that it does go on much longer 
than it would on a rough surface, and that a well-lubricated wheel revolves 
longer than one which is dry. Hence they concluded that friction or adhesive 
contact with matter is the chief circumstance which limits the principle. 
Since the same push will propel an object over a longer distance in air than 
in water or in contact with a solid surface, they argued correctly that this 
limitation is usually unimportant when the matter in contact with the moving 
body is rarefied. They also recognized that air friction may be considerable 
if the surface of an object is large compared with its bulk, and Newton himself 
carried out an important experiment to find out how far air friction can be 
neglected (see p. 376). For calculating the range and height of the slow- 
moving short-range projectiles of Galileo's time, the error due to air friction 
is very small. 

It was thus natural to go a step farther. If there were no air, and no contact 
with matter, there would be nothing to stop a cannon ball from going on in 
a straight line at the speed with which it left the muzzle, except the fact 
that like all other solid bodies it is being pulled earthwards during its range. 
If so, why should not the planets mote in a straight line through the vastness 
of empty space with no friction to stop them? Might not the answer be that 
they were pulled sunwards as the cannon ball in its curved path (Fig. 171) is 
pulled earthwards? This was the problem to which Newton's first "law" was 
preliminary. The words in which he propounded the principle of inertia, 
though seemingly speculative in circumstances with which Newton was not 
concerned, are easily intelligible in their own setting. 

MECHANICS IN THE ANCIENT WORLD 

To trace the steps which led to Newton's mechanical view of the solar 
system we must start with machines as the word was still used in Newton's 
time. With the exception of the sailing ship, the windmill, and the water 



Wheel) Weight^ and Watchspring 243 

wheel, machines were still devices for spending human effort in a more 
convenient way. A machine was not something that did work for you like 
a dynamo. It was something which gave you mechanical advantage, another 
term for adapting a task to the normal rate at which a human being works 
efficiently or, more simply, allowing you to take time over your job. In effect 
they gave you the choice of carrying a light load over a long stretch instead 
of straining yourself with a heavy load over a short distance. It is sometimes 
essential to choose the first alternative, because there is a limit to the load 
against which the human muscles will contract at all. It is also a social 
limitation imposed by the circumstances of slave labour. An overseer can 



r> 



r-l-i 





d ~ 

FIG. 155. THE LEVER 

keep slaves on the move. Only freemen are capable of sustained maximum 
effort. While slaves were abundant the mechanical ingenuity of the ancient 
world was mainly focussed on devices of this sort, and what mechanical 
principles were recognized as such were chiefly concerned with them. 

The two broad generalizations of mechanics in the ancient world arose 
respectively on the one hand from the practice of architecture, navigation, 
and mining (stresses, lifting stone blocks, cargoes, and ore), and on the other 
from irrigation (water level) and assay. The principle of the lever and the 
principle of buoyancy are both set forth in the works of Archimedes, and 
circumscribe the theoretical basis of mechanics in Alexandrian civilization. 
The bearing of the latter on other problems which arose in artillery warfare 
will be touched on in Chapter VIII. The principle of the lever, which can 
be used to measure the mechanical advantage of the "simple machines" 
referred to in the last paragraph, in all probability emerged directly from 
the practice of warfare. Powerful catapults were first used in the campaigns 



244 



Science for the Citizen 

~~ ~1 2 T" 



4- 3 2 " I 




3th.3t4A.-fhm 

balance 



3ft. 




FIG. 156. Two WAYS OF USING THE LEVER PRINCIPLE 

of Alexander and became an important item of military equipment in the 
hands of his Roman successors. Archimedes who in addition to his 
other varied mechanical interests occupied part of his time in devising 
catapults,* was probably led to investigate the properties of levers to account 
for their action. 

* Archimedes' part in the defence of Syracuse in 2 15-2 14 B.C. is described by 
Polybius in the following passage from his Histories : 

"The city was strong. . . . Taking advantage of this, Archimedes had constructed 
such defence . . . that the garrison would have everything at hand which they might 
require at any moment. . . . The attack was begun by Appius bringing his pent 
houses, and scaling ladders, and attempting to fix the latter against that part of the 
wall which abuts on Hexapylus towards the east. At the same time Marcus Claudius 
Marcellus with sixty quinqueremes was making a descent upon Achradina. Each of 
these vessels was full of men with bows and slings and javelins. . . . On these double 
vessels, rowed by the outer oars of each of the pair, they brought up under the walls 
some engines called 'Sambucae* [large scaling ladders for use from ships]. . . . But 
Archimedes had constructed catapults to suit every range; and as the ships sailing 
up were still at a considerable distance, he wounded the enemy with stones and darts, 
from the tighter wound and longer engines, as to harass and perplex them to the last 
degree; and when these began to carry over their heads, he used smaller engines 
graduated according to the range required from time to time. ... As often too as they 
tried to work their Sambucae, he had engines ready all along the walls, not visible at 
other times, but which suddenly reared themselves above the wall from inside, when 
the moment for their use had come, and stretched their beams far over the battlements, 
some of them carrying stones weighing as much as ten talents, and others great masses 
of lead. So whenever the Sambucae were approaching these beams swung round on 
their pivot the required distance, and by means of a rope running through a pulley 
dropped the stone upon the Sambucae, with the result that it not only smashed the 



Wheel, Weight, and Watchspring 245 



--- t/sru. > 



45 Ib. beam, 



22| Ib. 




Total 

J '2 ft, on this side, 



A*rn 



FIG. 157. THE LEVER 



= 45ft-lb. 
AVERAGING Our THE WEIGHT OF THE BEAM 



The Law of the Lever states the distances from the fulcrum at which 
different weights must be attached if they are to balance. The rule given by 
Archimedes is that the load is inversely proportional to the distance. That is 
to say, two weights, W at a distance D, and w at a distance d from the fulcrum 
balance when* : WD = wd. 

machine itself to pieces, but put the ship and all on board into the most serious danger 
Other machines which he invented were directed against storming parties. . . . 
Against these he either shot stones big enough to drive the marines from the prow; 
or let down an iron hand swung on a chain, by which the man who guided the crane, 
having fastened on some part of the prow where he could get a hold, pressed down 
the lever of the machine inside the wall; and when he had thus lifted the prow and 
made the vessel rest upright on its stern, he fastened the lever of his machine so that 
it could not be moved; and then suddenly slackened the hand and chain by means of a 
rope and pulley. . . . Such was the end of the attempt at storming Syracuse by 
sea. . . . So true it is that one man and one intellect, properly qualified for the par- 
ticular undertaking, is a host in itself. ..." 

* In this equation all the quantities are positive. It is, however, sometimes con- 
venient to represent co-ordinates (i.e. distances with a sign attached) to the left of 
the pivot as negative, whilst those to the right are still considered positive (or vice 
verso). If this is done, D will be a negative quantity, so that we must write the equation 
as WD = wd. Though the left-hand side of this equation has a negative sign 
attached, it is really positive. The quantity wd measures the turning-power of the 
weight w y called its moment^ about the pivot. In our example the moment is positive 
if it tends to turn the beam clockwise about the pivot and negative if it tends to turn 
it anti-clockwise, so it is evident that WD should be a negative quantity. 



246 



Science for the Citizen 



A familiar application of the principle in the everyday life of our own time 
is the type of scales used on railway stations for weighing luggage, or in 
hospitals and chemists' shops for weighing the human body (Fig. 156). Its 
bearing on the catapult is interesting, because it involves an important 
consequence which was not fully grasped till the displacement of human 
effort by machinery made it necessary to measure work. We all know that 
practically no effort is required to tilt the pans when two weights are balanced 
on a scale. If the pulling power or Force which your own arm can exert is 
capable of raising a hundredweight, it can also balance a ton at the end of a 




FIG. 158. WEIGHING MACHINE BASED ON THE PRINCIPLE OF THE LEVER 

lever, when the hand is applied to the opposite end at twenty times the 
distance of the load from the fulcrum. A negligible additional effort is then 
required to jerk the load upwards. 

Not being concerned with machines which are driven without human 
effort, Archimedes did not recognize that his principle gives us a clue to 
the right way of measuring work, so that we know when we are working 
economically. If you think of one of the earliest mechanisms devised to 
facilitate human labour, you will see that the modern way of measuring 
work is a matter of common sense. The first Magdalenian man who drew a 
sledge, must have been fully aware that a day's work could be reckoned by 
the pull exerted and the distance covered. Using E for work, p for pull and 
h for distance, we should say : 



x h 



Wheel, Weight^ and Watchspring 247 

If we call a unit of work, work done in traversing a unit of distance with a unit 
of pull 

E = pA 

The law of the lever tells us that this way of measuring work in so many 
Ib. wt. and so many feet (foot pounds weight) is in general a satisfactory way 
of measuring work. When two weights are balanced 

WD = wd 




H! 




FIG. 159. THE PULLEY 



If we now tilt the beam a little, so that the small weight descends through 
h y and the large ascends through H., Fig. 155 shows that 



It follows, therefore, that 



D H 

WH = mh 



i.e. one load gains in power to do work to the same extent as the other loses 
power to do work. 

So if we measure work by the product of the weight lifted and the distance 
covered, no work is done in tilting a lever when it is just balanced. That 
we know this to be true shows that this is a satisfactory way of measuring 
the work of a machine as well as the day's work of Magdalenian man. 

The same principle can be applied to another ancient device, balancing 
weights by means of a pulley. If a cord is suspended over a simple pulley 
shown in the left of Fig. 159, two equivalent weights attached to its ends 
balance. If a weight is attached to the free end of the cord in a compound 
pulley like that shown on the right, experiment shows that it will balance two 



248 Science for the Citizen 

equivalent weights attached to the second pulley. The principle of the lever 
leads to the same conclusion as experiment. If the smaller one (w) falls 
through a small distance + h> the pulley is raised through a distance + \h. 
With the usual convention of signs, it therefore falls through |A. If no 
work is required, the total work done in a small shift is : 

wh - JWA = 



Yet a third "simple machine" for getting the same task done in a more 
leisurely way without reducing the total effort expended was the "screw and 
cogwheel," the name of which is self-explanatory. In a work which has 
come down to us in an Arabic translation, Hero, who was the most volu- 
minous writer on mechanics in the Alexandrian age, describes a hypothetical 
combination of lever and multiple pulley with screw and cogwheel for raising 
weights. How scant was the encouragement for mechanical ingenuity in the 
modern sense is illustrated by another remarkable device which is described 
in his Pneumatica. The expansion of air in a hollow altar during the sacrificial 
fire is made to force water out of a closed vessel into buckets. The descent 
of the latter, when full, is used to pull open the doors of the temple where 
profane labour was precluded. 



WEIGHT AND MASS 

During the fifteenth century the elementary principles of balancing 
weights as they had been expounded in the Alexandrian schools became 
known in Italy, especially through a treatise of Leonardo da Vinci, who 
described his own experiments on gliders and a hypothetical man-driven 
aeroplane. What proved to be an important bridge between the mechanics of 
the old world and modern science was the extension of the Archimedean 
principle to one of the oldest devices for producing mechanical advantage. 
Stevinus of Bruges, a quartermaster in the army of William of Orange, 
carried out experiments on the equilibrium of a load hanging vertically with 
a load resting on a smooth slope. Like Archimedes, Stevinus attempted un- 
successfully to improve the numeral system of his time, and like Archimedes 
he was specially interested in the mechanics of warfare and navigation. He 
designed "machines" for lifting the Dutch fishing boats above high-water 
mark, and was an expert in the art of fortification. 

In applying the principle of the lever to other situations, it is necessary 
to add a very important qualification, which has not been clearly stated 
so far. In all the cases we have so far considered, the body has moved in the 
direction in which the force applied acts, either forwards, giving positive work 
which we do ourselves, or backwards, giving negative work which is done for 
us. For the sledge, the direction was horizontal, and for the lever and compound 
pulley, vertical. If we roll a truck up several hills, we soon find out that more 
work is needed to push it up a steep hill than is needed to push it through 
the same distance up an easier slope. If we forget about frictional resistance, 
the amount of work needed depends only on the total gain in height above the 
original level. 



Wheel) Weight^ and Watchspring 249 

This suggests the law of balancing weights when one (w) hangs vertically 3 
and the other (W) rests on a smooth slope inclined at a fixed angle (B) to 
the horizontal. If the weight w descends through a vertical distance D in 
Fig. 160^ the weight W ascends through the same distance D along the inclined 
surface. Since sin B = h - D> its vertical displacement h = D sin B. If 
we measure the work done on W in rising through the vertical distance h 
and the work done by w in falling through the vertical distance D as we have 
done in the treatment of the lever and pulley (i.e. by the products of the 




FIG. 160. REST (BALANCE OF Two OPPOSED MOTIONS) ON THE INCLINED PLANE 

The Law of the Inclined Plane is that if a weight found to be w times some standard 
weight by the balance when hanging vertically just supports a weight W times the same 
standard lying on an inclined surface of elevation B, 



w = W sin B 

Weight hanging vertically 
Balanced weight on slope 



- sinB 



weights and their corresponding vertical displacements) the total work done 
by both weights is W/z + wD. So if no work is done in shifting them when 
the weights are balanced 

W/z + wD = 

W/* = wD 
.'. WD sin B = wD 

w = W sin B 



250 Science for the Citizen 

According to this the weight required to balance a 10 Ib. load resting on 
an inclined plane of 30, when it hangs vertically is 

10 x sin 30= 10 X 0-5 Ib. 

i.e. 5 Ib. If the angle of the sloping surface is 45 the weight required is 
10 x sin 45= 10 X 0-707 Ib. 

i.e. just over 7 Ib. The hypothesis is evidently true at the limits. If the slope 
is 90 the system is equivalent to the simple pulley of Fig. 159, and the 
weight required is 

10 x sin 90 = 10 Ib. 

The law of the inclined plane raised a distinction which had never been 
clearly disentangled in the mechanics of antiquity. For the purposes of 
metallurgy or merchandise, two weights are equivalent when they balance 
one another on a pair of scales, i.e. when they are in equivalent positions 
relative to the fulcrum. As far as their pulling power is concerned they are 
only equivalent when this condition is realized. In speaking of weights, we 
must therefore distinguish between mass which we "weigh" on a pair of 
scales and pulling power or force which depends on something else.* The 
mechanics of antiquity stopped short at identifying this something else with 
the position in which a body was placed when it balanced another body at 
rest. This provided a satisfactory explanation of what you need to know in 
designing the best Archimedean catapult. It did not throw any light on the 
propelling power behind a cannon ball, or the pull of the earth's magnetism 
on the mariner's compass needle. In Alexandrian mechanics pulling power 
or force always implied some tangible link, a common surface on which two 
weights rested, or a string connecting them. The phenomena of magnetism, 
as set forth in Gilbert's De Magnete (1600), defied all the accepted conventions. 
Here was pulling power with no support and no attachment. 

FORCE AND MOTION 

The experiments of Stevinus suggested the clue to a new technique of 
measurement. Besides being able to balance another weight at rest in virtue 
of its mass and position, a weight can also generate motion, as when a load 
attached to a cord wound about the axle of a wheel is allowed to descend. 
We all know that we slide down a slope more quickly if it is steep; that is, 
the weight of our bodies can generate more motion if the angle of descent is 
increased. So, too, the balancing power of a weight also increases if the 
elevation of the slope on which it rests is increased. Hence the principle of 
the inclined plane suggests two new possibilities. One is that the pulling 
power of a load in different positions simply depends on the fact that its 
power to generate motion also depends on its position. The other is that two 
weights balance when their power to generate motion is equally great and 
is exercised in opposite directions. The obstacle to pursuing this plausible 
clue in the sixteenth century was the fact that no one had yet attempted to 
measure the way in which things fall. According to the Aristotelian world 

* See also p. 292. 



Wheel, Weight, and Watchspring 251 

outlook, rest was the rule and motion an exceptional state of affairs. The 
alternative view that rest is only motion at a deadlock had little to commend 
it in the ancient world. Before the introduction of gunpowder there was no 
pressing social necessity to compel anyone to make exact measurements of 
the rates at which bodies fall. 

Indeed, it is not likely that Aristotle's doctrine would have been contested, 
if range-finding had not come to the aid of curiosity in an age when the fate of 
a new civilization depended on a new technique of war. Aristotle had taught 
that a "body is heavier than another which in an equal bulk moves downward 
more quickly." The belief that the vertical motion of a body is proportional 
to its density (mass per unit volume) is not intrinsically unlikely. Our common 
experience of falling objects seems to confirm it. We can see leaves falling 
from a tree, or a cup falling when it leaves our hand, but we cannot see a 
cannon ball in its course. We have blown up paper bags or toy balloons as 
children, and bladders inflated with air were part of the medieval jester's 
equipment. The behaviour of all these familiar objects seems to agree with 
Aristotle's teaching, and the simple discovery which enabled Galileo to lay 
the foundations of scientific artillery therefore ranks as one of the most 
daring achievements in science. 

Apart from the last passage cited, Aristotle had not committed himself 
to any very definite statement about the way in which things fall. His dis- 
cussion of physical problems, like his political philosophy, is mainly con- 
cerned with justifying, defending, and making a lawyer's case in favour of, 
the world as it is. In his doctrine of the state every individual had a fixed 
social class which was right and proper in the nature of things. In his natural 
philosophy every material object also had its proper place; and if it got out of 
its proper place, nature personified as the strong arm of the law put it back 
sooner or later. Unhappily for his influence as a philosopher Aristotle com- 
mitted himself to one assertion which is easy to test by dropping a cannon 
ball and a croquet ball of the same size from the top of a building. Although 
it is obviously true that some very light things like toy balloons and some 
objects with a large surface relative to bulk, like feathers, leaves, or wood 
cut in fine shavings, do fall more slowly than cannon balls, it is also true 
that most compact objects fall from the same height in approximately the 
same time. That this is so within wide limits of relative heaviness was the 
surprising new truth which Galileo established. 

Mach says that Galileo's outlook was essentially modern, because he 
asked how things fall instead of why they fall. There is nothing specifically 
modern in this except in so far as the ceremonial and priestly outlook is 
losing ground, while the attitude of the artisan exercises an increasing influ- 
ence on our social culture. Galileo was modern only because he was work- 
manlike. The history of science has been a long struggle between the same 
two conflicting inclinations. Priestly and ceremonial speculation is content 
to reflect on the world, or attempts to propitiate an unseen supposititious 
purpose in things personified explicitly as deities or implicitly in the "explana- 
tions" of the unofficial theologians who teach philosophy. If you follow to 
its logical conclusion Aristotle's doctrine that everything has a place to which 
it belongs you end in an act of special creation. Any question which begins 



252 Science for the Citizen 

with why leads you back to an assumed design by an individual who cannot 
be identified but may possibly be bribed. The artisan or technician looks 
at things as they are, and finds out how they can be changed in accordance 
with human needs. The influence of their world view expands as the satis- 
faction of man's common needs becomes the common business of mankind. 
A workmanlike or truly scientific explanation gives you a recipe for doing 
something. The distinction between a real or scientific explanation and a 
philosophical one is easier to see nowadays because few people believe that 
prayer can control meteorological events. So long as they did, teleological 
"explanations" were intelligible. 

A cannon ball or a croquet ball dropped from the top of a three-story 
building, say 36 feet high, reaches the ground in about a second and a half. 
Even from the top of the highest New York skyscraper the drop would be 
over inside nine seconds. It goes without saying that direct measurement 
of how bodies fall vertically cannot be precise, unless there are very good 
recording instruments to use. Stop-watches, electric signals, metronomes, 
kymographs (Fig. 196), had not been invented in Galileo's day. He had to 
tackle the problem by an indirect method which involved two kinds of 
measurement. He first established a simple rule connecting the distance (d) 
traversed and the time (f) of the descent of a ball (see Figs. 161 and 162) 
on a smooth gentle-sloping surface. Then he sought for a rule connecting 
the time taken to traverse a given distance with the angle at which the surface 
is tilted. The limiting angle is 90, when the ball falls vertically. 

The time taken for a ball to descend from different levels on a gently 
inclined surface is not difficult to measure. Having no stop-watch, Galileo 
let water drip steadily into a vessel during the interval occupied by the descent, 
and weighed the water to get an estimate of the time. A table of results 
obtained in an experiment of this kind discloses a simple numerical rule 
which is at once recognized when a graph of the distances traversed and 
time taken is plotted. The graph answers to the textbook parabola y =-- px 2 , 
and an examination of the figures shows that the ratio of the distance traversed 
to the square of the time taken is fixed, i.e. 

d 

p = k or d =r kt*. 

If we now look at the motion of the ball from a different point of view, 
there is more in this than meets the eye at a first glance. Galileo did so. He 
introduced what is now a familiar but was then an entirely novel criterion of 
motion. Crude estimates of speed (or velocity as we shall now say when 
speaking of motion in a particular direction along a straight course) had 
probably been made from the first experience of travel, route marches, and 
public games. Hitherto there had been no occasion to measure how motion 
grows. Galileo adopted the modern measure of acceleration, or velocity gained 
in unit time. There is no longer an unusual flavour in the word, though we 
shall see later that it is used loosely in everyday speech and that the terms 
speed and velocity, used interchangeably when we are describing motion 
in a straight path, have to be restricted when we are describing motion 



Wheel, Weight, and Watchspring 253 

round a bend. For the present we need only concern ourselves with the 
ordinary use of the term. If the speedometer reading of a car increases 
steadily during 10 seconds of a straight course from 30 miles per hour 




2-24 



2-50 



FIG. 161. A SMOOTH BALL RUNNING DOWN A VERY SMOOTH SLOPE AT 30 

UNDER GRAVITY 

If acceleration is constant, the acceleration calculated from the formula d lat z 
will be the same whether the distance (d) fallen is large or small. The figures in the 
horizontal row give the total interval of time from the start at each 10-foot stage 
in a 50-foot journey. The results tabulated as follows give a numerical illustration 
of the reasoning in the text. The velocity gained per second, i.e. the acceleration, 
at different stages in the descent is given to the nearest unit, and does not vary 
consistently in either direction from an approximately constant figure of 16. 
Dis- 



tance 

in Time in 
Feet sees. 



Velocity 



at 
time 



Velocity 
gained 



Velocity 

gained per 

second 



10 



30 



40 



s 93 



1-94 J 



i-58 i 12 



10 



(2 1 74 - 8 93) (1 35 -0-56) 1281-0-79 
- 12 -SI - 70 - 16 



35 



U27 78-21 74^) (1 76-1 35) 
f -6 04 -0 41 



i- 'J 24 - 1 94 



10 



6-04-0-41 
=15 

Ui'94-f 1 58) J 
=- 1-76 \ 

((33 33-27 78) (2 09-1-76) 5 55-fO-33. 
f - 5-55 -0 33 =17 

.',(2 24 -H 04) J 
2 09 



l y_ 38 46 



>2 :>() -2 24 



K2 504-2 24) 
2 37 



(38-46-33-33) (237-2-09) 513-0-28 
I -5-13 -0-28 ^18 



(or 44 feet per second) to 45 miles per hour (or 66 feet per second) it has 
gained in velocity 22 feet per second in 10 seconds or 22 feet per second 
in 1 second. Its mean acceleration is -f 2-2 feet per second per second. 



254 



Science for the Citizen 



If the reading falls from 45 to 30 miles per hour,, it has lost 2-2 feet per 
second in one second. Its mean acceleration is 2-2 feet per second per 
second.* 

If a thing moves with a changing velocity., its mean velocity is the fixed 
velocity at which it would have to move in order to cover the same distance 
(d) in the same time (f). Thus its mean velocity is d ~- 1. If it gains the same 
velocity in equal intervals of time., this is half the sum of its initial and final 
velocities. So if it starts from rest (i.e. zero velocity) and has reached a 
velocity v at the end of t seconds, this is i(0 + v) = i v. 



2d 



2 ft. 




13 -ft. 



Seconds J 4 ^ $4 1 1\ 1*2 

FIG. 162. MOTION ON THE INCLINED PLANE 

The final velocity v is then the velocity it has gained in the time t. Since its 
acceleration (a) is the velocity gained in one second, 

v 2d 



If it moves from rest with a constant acceleration., the ratio of the distance to 
the square of the time taken is therefore fixed. Galileo's experiments showed 
that when a ball rolls down a smooth slope, the distance traversed bears a 
fixed ratio to the square of the time occupied by the descent. They therefore 
showed that the ball rolls down the slope with a constant acceleration. 

So long as the slope is the same, this is true within wide limits for the 
acceleration of balls of different weights and densities. The next thing is to 

* "Per second per second" is often written "per sec. 2 ' 2 . 



Wheel, Weight^ and Watchspring 255 

find how the slope, i.e. the angle b which the surface makes with the hori- 
zontal,, affects the acceleration. This is done by varying the slope and 
measuring the time taken to traverse a fixed distance. We get a clue to the 
kind of relation which exists from the fact that a body moves most quickly 
when it falls straight down (i.e. b == 90), and remains still when the surface 
is quite flat (i.e. b = 0). So the acceleration depends on something about 
an angle which is greatest when the angle is 90, and is zero when the angle 
itself is zero. This is true of the sine of an angle., and experiment shows that 
the acceleration is directly proportional to the sine of the angle of slope, i.e. 

a oc sin b 
or putting it in the form of an equation, in which g is a constant 

a g sin b 

Experiment shows that if the slope is 30 the acceleration is approximately 
16 feet per second per second, i.e. the velocity along the slope increases 
every second by 16 feet per second. So 

16 - g sin 30 
.'. 16 = X i 
.*. g 32 (approximately) 

Since sin 45 =-- 707, if the slope were 45, the acceleration would be 

32 x 0-707 = 22-6 feet per second per second. 
If the angle b is 90 the body falls vertically (Fig. 163), and since sin 90 1 



i.e. g is the acceleration with which a heavy body falls vertically. Galileo 
showed this was so by finding the time taken by heavy objects to fall from 
the top of a tower to test the rule in the limiting case. The acceleration can 
be found as before by using the rule 

2d 



So if the rule holds good, when d is the vertical height of the tower, and t 
the time between dropping the body and seeing it strike the ground, 



= I6t z (approximately) 

Thus a pebble dropped from the top of a lighthouse 100 feet high would 
strike the water A/100 -f- 16 =. 2| seconds later. 

In this way Galileo established two conclusions which Newton used in his 
theory of universal gravitation. The first, called the law of terrestrial gravita- 
tion, is that when bodies fall vertically (i.e. along the plumb line which points 
towards the centre of the earth) they move with a constant acceleration. Near 



256 



Science for the Citizen 



the earth's surface this is approximately 32 feet per sec. 2 . In air it is inde- 
pendent of mass or density within wide limits. The second conclusion is that 




FIG. 163. MOTION DOWN AN INCLINED PLANE 



The law of Motion is : 



Acceleration (A) along slope 
Acceleration (a) of free vertical fall 



= sin b 



= sin b 



The law of rest (F?. 1 60) is : 

Vertically suspended mass (m) 
Balanced mass on slope (M) 
Hence 

A a = m -~ M 

MA ma 

or, in this case, MA = mg. 

Thus the pulling powers exerted by two masses in different situations are equal when 
the product of one mass and the acceleration with which it would move if not con- 
strained by the second is the same as the product of the second mass and the accelera- 
tion with which it would move if it were not posed against the first. 

the law of equilibrium of solid bodies at rest is only another way of stating 
the same law of motion. The mechanics of antiquity had shown that equili- 
brium of solid bodies at rest involves (p. 250) position as well as mass. The 



Wheel, Weight^ and Watchspring 257 

inclined plane suggested that this might be due to the fact that in different 
positions heavy bodies do not always have the same power to generate motion. 
Galileo's experiments show us how the position of a body affects its motion, 
when it is displaced. The law which describes how weights balance one 
another when one of mass M rests on a smooth slope at an inclination b to 
the ground level, and the other of mass m hangs vertically, may be written 

m . 
= sin b 

M 

The law of motion down a smooth slope is 

- = sin b 
g 
Comparing both, we see that 

m a 

M = 
mg = Ma 

Thus our first suspicion is correct. Two loads balance on a smooth flat 
slope when the product of the mass of one and the motion it would generate 
if released is the same as the product of the mass of the other and the motion 
it would generate if released. In other words, they remain at rest because 
their power to generate motion is equal and opposite. Hence the mass-accelera- 
tion product is a way of measuring the pulling power (F) of a load, i.e. 
F m . a. If one body is falling with a smaller acceleration than another 
it has less effective pulling power, unless its mass is proportionately larger. 
To hold a greater mass of one demands a greater mobility of the other. 
Thus a force has two aspects, one acceleration which measures the mobility 
of an object, the other mass which measures its sluggishness or inertia. 

To measure forces by the mass-acceleration product we have now to 
agree about the measurement of mass and of acceleration. When the unit of 
mass is 1 Ib, the unit of distance 1 foot, and the unit of time 1 second, the 
unit of force is called the poundal, i.e. one poundal is the force required to 
impart an acceleration of 1 foot per second per second to a mass of 1 Ib. 
The force exerted by a 1 Ib. "weight" falling under gravity or the force 
required to prevent a pound "weight" from falling under gravity near the 
earth's surface is approximately 32 poundals. Since g = 32 (approximately), 
m = 3f2 Ib. when mg = 1. Hence one poundal is the pulling power of a 
half ounce weight when suspended in a vertical position. It is therefore the 
pulling power of a spring balance when the scale reading is oz. 

In the international metric system based on the reports originally drawn 
up by the French Academy for the National Assembly of the French Revolu- 
tion, the units of mass and length are respectively the gram and the metre. 
The former is conveniently chosen as the mass of 1 cubic centimetre of 
water at 4 C. (i.e. the density of water in grams per c.c. is 1 at this tem- 
perature, when its density is greatest). The unit of force, the dyne, is the 
pulling power which can make 1 gram move with an acceleration of 1 cm. per 
second per second. Since 1 British foot = 30*48 cm., and 1 British pound 

i 



258 Science for the Citizen 

453-6 grams, 1 poundal is equivalent to 30-48 x 453-6 = 13,826 dynes. 
Since 32 feet == 975 cm., 1 gram hanging vertically exerts a pull of 975 dynes, 
and 1 Ib. exerts a pull of 975 x 453-6 = 442,260 dynes at a place where the 
value of g is exactly 32 feet per second per second. We shall see later that the 
value of g is generally a little larger than 32 feet per second per second, and 
also varies a little with latitude and altitude, being at sea-level 32-09 feet or 
978 cm. per second per second on the equator and 32-26 feet or 983 cm. 
per second per second at the North Pole. Fifteen miles above sea-level at the 
equator, it is about 30*9 feet or 941 cm. per second per second. 

Before we go on, four simple rules which connect time, distance in a 
straight path, velocity, and acceleration when the acceleration is constant, 
should be committed to memory, so that you recognize them at once, when 
you come across them in what follows. They are as follows : 

(1) If an object moves through a distance d in a time t with a velocity v* 
since v -= d -f- t y 



(2) If an object is moving with a velocity v a at the beginning of an interval 
of time t and is moving with a velocity v b at the end of it, its mean velocity, 
which is the velocity with which it is moving half-way through the time 
interval, is l(v a + z; 6 ), and the distance traversed in the interval t is the same 
as if it moved at its mean velocity throughout the whole of it, i.e. 



(3) If an object gains a velocity a feet per second in each second, a is its 
acceleration; and it will have increased its velocity by at in t seconds. If its 
velocity was v a at the beginning of the interval t its velocity at the end of it 
will be Va. + at, so that its mean velocity is ^\v a + (v a + at}] = v a + \at. 
and the distance moved during the interval by rule (2) is (v a + \af)t. If the 
object starts from rest, so that v a = 0, 

(or a = 2d + * 2 ) 



(4) The fourth rule which has not been used so far is illustrated in Fig. 164 
and explained in the legend. If an object moves with a velocity v and an 
acceleration a along a line (AC in the figure) inclined at an angle b to a second 
line (AB), an observer, who keeps his eye at right angles to the second line, 
watching its progress along the latter will see it move with a velocity v cos b 
and an acceleration a cos b. Ifa x is the acceleration referred to a line inclined 
at b to the path along which an object moves with acceleration a, 

a r = a cos b 

The first, third, and fourth should be memorized and tested with the aid of 
numerical illustrations and scale diagrams like Fig. 164. The velocity formulae 
should be tested from the numerical data in the boat diagram of Fig. 167. 

The last rule throws a new light on the measurement of force if we put 
it in a more active form. The same source of motive power which propels a 
body with an acceleration a in one line gives it an acceleration a cos b along 



Wheel, Weight, and Watchspring 259 



server 

s obizcb's' -progress' 
i *i_ J / 7* J? ^ /* 
along its actual path, AC 




'W 



^ 



s* 



'^, 



after 2 sett (30 ft) 

Second 
> r 74 "seiT Observer 




Third Observer 
watches it alcm& 

FIG. 164. TRIANGLE OF VELOCITIES AND ACCELERATIONS 

This illustrates the rule that if a body moves with a velocity v or an acceleration a 
along a straight path, its velocity v x and its acceleration a x along a line of reference 
inclined at an angle b to the path can be found by using the formulae : 

D X = v cos b 
a x a cos b 

The object shown moves from rest at A with an acceleration a (here 25) feet per 
second per second for t (here 2) seconds along AC. Its motion referred to the line 
AB is its motion as it would be seen by an observer looking along a line at right angles 
to AB at each stage in its course. This would be equivalent to watching its shadow 
projected on a ground glass screen at AB with a beam at right angles to the screen. 
Thus AB, AC, and the observer's sight line form a succession of right-angled triangles 
in which cos b = AB 4- AC. Its acceleration along AC is 2AC 4- r 2 , and its acceleration 
(a*) along AB is 2AB 4- t z . Hence ax 4- a = (2AB 4- r 2 ) 4- (2AC -T- r 2 ) = AB 4- AC 
= cos b. At the end of t seconds its velocity (v) along AC is at and its velocity (vx) 
along AB is a x t. Hence v x 4- v a x 4- a = cos b. Along the line at right angles to 
AB a i.e. BC, inclined at 90 b to AC the acceleration (%) is a cos (90 b) = a sin b, 
and similarly v y = v sin b. In the figure the ratio of the sides of the triangle is 5 : 4 : 3, 
so that cos b = 0-8 and sin b= 0-6. The figure If is approximate in the 
More exactly it is V3. 



260 



Science for the Citizen 



a line inclined at b to the other. If a body falls freely under gravity with an 
acceleration a, its acceleration relative to a slope at b to its vertical line of 
fall (or c = (90 b) to the ground) is g cos b or g sin c (since sin (90 &) 
= cos b). So its acceleration when moving down a smooth slope is the accelera- 
tion referred to the same line of motion by the same motive power which 




^ Observer 

rec&i'dirug trie 
progress of 
the, ball 



to 



soe ..... 



op 



etc. 




FIG. 165. MOTION DOWN THE INCLINED PLANE AS FREE VERTICAL MOTION RESOLVED 

ALONG THE PLANE 

When a body falls vertically through a distance d (= %gt z ) its acceleration (#) under 
gravity is 2d -r r 2 . An observer (O) watching its progress along a line inclined at b 
to the vertical (or c = 90 b to the ground) would see it move through x = d cos & 
with an acceleration 2x -~- t 2 = g cos ? or sin c. This is the value to which actual 
acceleration down the slope approximates when friction is negligible, and hence its 
acceleration down the slope is equivalent to its acceleration when falling vertically 
resolved along its path. 

imparts an acceleration g in the vertical direction. The law of terrestrial 
gravitation may therefore be put in another way by saying that the mobility 
of falling bodies can be attributed to the same motive power which pulls 
them in the same line of action with a force proportional to their masses. 
Thus the force of gravity acting on a weight of M Ib. is 32M poundals a and 



Wheel, Weight, and Watchspring 261 

a force of 1 Ib. weight, i.e. the force with which a pound weight is pulled 
earthward, when it hangs from a spring balance, is 32 poundals. 

The view to which Galileo's experiments led therefore reduced Aristotle's 
logic to an absurdity. If the same motive power pulls all material bodies near 
the earth's surface in the direction of the plumb line, i.e. towards the earth's 
centre, the place to which all bodies "belong" is the centre of the earth. It 
is difficult to see how there can be enough room for them. Consequently 
there is no intelligible meaning in the question, where does a body "belong"? 
The only question for which we can hope to find an answer is: how do 
bodies actually behave? This we can only discover by observing them, 
making theories about how one observation is connected with another, and 
then testing them to see whether they are right. 

THE PATH OF THE CANNON BALL 

Artillery was at first used more to destroy the enemy's morale than for 
the material destruction it produced. To be effective in the latter sense it was 
necessary to know how to hit the objective. Accuracy of aim demanded a 
theoretical basis. Galileo's law of falling bodies provided it with one. A 
cannon ball when fired into the air has two movements. One results from 
the explosion, giving it an initial velocity in a certain direction depending 
on the angle which the muzzle makes with the horizon. The other movement 
is due to the fact that gravitation is pulling the projectile vertically down- 
wards, so that it tends to gather velocity towards the earth. The result is 
that it travels along a curved path. The theory of the pendulum clock, and 
with it that of the earth's orbital motion, depends upon understanding how 
movement in one direction can be represented as the result of movement in 
two other directions. 

It is so natural to split a complicated form of movement into simpler 
components that you may not have realized that we have done so already. 
The description of the sun's apparent course in the heavens as a diurnal 
rotation in a plane parallel to the equinoctial and an annual retreat in the 
ecliptic is a purely arbitrary trick to assist us (Fig. 166) in following its 
continuous apparent track through a closely wound closed spiral. The same 
trick was suggested by the practice of navigation in another way. To reach 
a destination when rowing or sailing in a steady current, you have to keep 
the boat's prow tilted away from the direct line of approach in the direction 
opposite to the current (Fig. 154). The velocity of currents is easy to measure 
by the progress of corks floating away from a vessel at anchor, and experience 
gives you a fairly good estimate of your average working velocity when rowing 
in still water. 

To find your progress along the actual line of approach you have only 
to lay off the distance (AB) a cork would drift with the current, and the 
distance (BC) you would move in still water during the same interval of time 
along the direction of the boat's axis. These two lines form the sides of a 
triangle of which the third (AC) is the distance covered in the same time along 
the actual path of motion, or, what comes to the same thing, the two adjacent 
sides of a parallelogram whose diagonal through A gives the distance of the 
actual path. If the interval of time is one unit, and the velocity is fixed, the 



262 



Science for the Citizen 



component and resultant distances are numerically the same as the correspond 
ing velocities. The same construction holds for accelerations. Since 
d = \at* = \a when t = 1, i.e. one unit of time after the motion starts, 
accelerations are directly proportional to distances traversed in equivalent 
time intervals from rest. 

In dealing with several simultaneous movements it is much better to 



ivr 

Diurnal v^-^ 




Single. 
Uoiwlz 



FIG. 166. SUN'S APPARENT MOTION IN THE CELESTIAL SPHERE 

The sun's apparent motion in the celestial sphere is continuous. Though we find it 
convenient to think of it as two separate motions, a diurnal rotation westwards over 
the horizon, and an annual retreat eastwards, a quick motion picture would reveal it 
as movement in a closed spiral with 365 turns. A closed spiral of five turns is here 
shown to simplify the issue. The figure would represent a continuous tableau of the 
sun's apparent motion through a year, if the earth took 73 of our days to complete a 
single revolution about its axis. 



follow the nautical plan by keeping a separate balance sheet of eastward 
(positive), westward (negative), or northward (positive), southward (negative), 
bearings, as explained in Figs. 168 and 169. The map method is really, as it 
is historically, the same thing as the graphical method of representing a 
movement, i.e. so many ^-measurements and so many simultaneous y- 
measurements. All you have to do is to add up all the east-west bearings 
or ^-measurements and all the north-south bearings or ^-measurements to 



Wheel) Weight., and Watchspring 263 

Bruige 




< ll_ ____ > 

Ship 3 nv3verw2Tit (x) 

FIG. 167. DISSECTION OF COMPLEX MOVEMENT 

During an interval t (here 3 ^ seconds) the boat moves with a fixed velocity 12 miles 
per hour (17 :t , feet per second) through a distance 56 feet along a straight canal, while 
the man moves straight across the deck 14 feet wide with a fixed velocity of 3 miles per 
hour (4r feet per second). Relative to an observer at right angles to the line AC the 
man appears to pursue a steady path with a velocity v = AC -h t. Since AC 2 = 
CD 2 + AD 2 



So if we call the man's velocity relative to the bridge v }J --- CD -f- t, and his velocity 
relative to the bank v x = AD t 

V 2 = v~ 4- vl 

His resultant velocity along the actual path which he pursues in space (shown in an 
air photograph with the boat blacked out) is therefore V9 -f 144, or approximately 
12 J miles per hour. Since tan b = CD -r AD = 3 ~ 12 or 0-25, the inclination of his 
path to the bank is the angle whose tangent is 0-25, approximately 14 from tables of 
tangents. Knowing the angle we could deduce his velocity relative to the bank (boat's 
motion) and to the bridge (his own) from the cinema record taken from the air, from 
the fact that he moves through CD relative to the bridge and AD relative to the bank 
during the time taken to move AC with his resultant velocity v. Since cos b = AD 
AC = (AD ~ t) ~ (AC -f- 1\ cos b}= Vx -r- v and v x = v cos b. Similarly v y = v smb. 



walks Jm.p 
(relative to boat) 
this- 




olups motion 

4-m.p.h. 
(relative to current) 



FIG. 168. RELATIVE MOTION IN THE SAME STRAIGHT LINE 

Distances traversed in the same time and hence velocities or accelerations can be added 
or subtracted according to direction. In the figure the resultant motion of the man 
relative to the bank is 4 2| 3 1J miles per hour, using the positive sign 
for motion left to right. 



264 



Science for the Citizen 



get the x and y measurements (E-W and N-S bearings) of the resulting 
path. If all the distances in a chart of this kind correspond to the same unit of 
time, the final result does equally for velocity or acceleration, since the velocity 
of a moving object after moving for t units of time from rest is proportional 
to the distance through which it has moved, and this is also proportional to 
the velocity it has gained, i.e. its acceleration. If the time interval from rest 
is 1 unit, velocity and acceleration are represented by the same number of 
units and the number of units of distance covered is half as great. 




FIG. 169. GRAPHICAL REPRESENTATION OF COMPOSITE MOTION 

In the graphical method of making a scale diagram to find the direction or magnitude 
of the resultant of several independent motions, we make use of the fact that motions 
along the same straight line are additive with due regard to sign. Each component 
motion is therefore split up into two at right angles like the N-S (;y measurements) 
and E-W (x measurements) bearings of a ship's moving course (which suggested the 
graphical method in the age when maps in latitude and longitude were first used 
extensively). We then add up (with due regard to sign) all the northerly and all the 
westerly items in the separate balance sheets of bearings, keeping the positive sign for 
the direction from S to N or from W to E and the negative sign for the converse. 

For example in (a), the motion OP can be split up into x and y 3 and the motion OQ 
split up into X and Y. Then by adding the E-W bearings together and the N-S 
bearings together we are able to find the direction and magnitude of the resultant 
velocity OR. 

Case (b) is similar, except that two of the bearings happen to be negative. 

In practice you will usually find it simpler to work in distances and use 
the graphical method of dissection which we shall first apply to an analogous 
but more pacific problem than the one which exercised Galileo. When a 
mail bag is dropped from an aeroplane it executes a path like a jet of water 
projected horizontally from a hose, the reason being the same. Its own 
inertia in virtue of the motion of the plane, which we shall assume to be 
moving parallel to the ground, gives it a drift forwards as it falls. So in order 
to know where to let go if it is to reach its proper destination it is necessary 



Wheel, Weight, and Watchspring 265 

to drop it before the aeroplane is directly over its destination, and to be able 
to calculate the distance between the position of the aeroplane at the correct 
time of dropping the mail bag and a point directly above the place where the 
bag is supposed to land. 

The graphical representation of the descent of a mail bag dropped from an 
aeroplane moving horizontally with a velocity v at a height h in Fig. 170 is 
made by combining the principle of inertia with what we know about the 
way in which bodies fall. If the aeroplane moves through x feet in t seconds, 
x = vt. The mail bag would continue to move in this way if its motion were 





AeropLeuw, with, TwrLzonaL velocity v wjjvdS x(~vt] 




^\ {dot in t sees. 


64-O 


(V--3MisSj\ 


512 


**-^^ i ~ti 


^-w 






% J 
A"^ 


J^ 










\ 




384 






\ 










\ 










\ t 










\ ? 


i 












256 


3 


T 


\ 










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128 






V 










\ 










\ 






1 1 > 


. \ 


( 


128 256 384 512 64O Vnp 



FIG. 170. DESCENT OF A MAIL BAG FROM AN AEROPLANE CRUISING AT 60 MILES 
PER HOUR AT A HEIGHT OF 040 FEET ABOVE GROUND. 

Air resistance is neglected, if the speed of the 'plane is not very great. 

not also affected by gravity. While it is moving through vt feet in the 
horizontal direction in virtue of its inertia, it is also dropping through a 
vertical height d = Igt 2 under gravity. If it sinks to a height y, while it 
moves forwards through a distance x, you will see from the figure that 



And since x = vt. 



and 



266 Science for the Citizen 

When the mail bag reaches the ground y and so 



Time 
(tecs ) 



2 

128 



= vV2h -:- 

3 
144 



(feet) 




Distance . . 64 

<"LTI jfeet) 



728 







\></ 
^p^ 



320 




a, = x-lr d :. x 
h-^-d .". h = 
FIG. 171. GALILEO'S CANNON BALL PROBLEM 
This is calculated for a muzzle velocity of 1 1 6 feet per second at an elevation of 56^. 

The numerical dimensions of the figure correspond to the course of a mail 
bag dropped from an aeroplane cruising 640 feet above ground at 60 miles per 
hour (88 feet per second). Hence the horizontal distance which it travels 
before it reaches the ground is 



881/1,280 4- 32 = 556 feet (approximately). 
That is to say, the bag must be dropped when the aeroplane has still to travel 



Wheel, Weight, and Watchspring 267 

556 feet before it is directly above the place of delivery. The points on the 
curve are plotted from the equation, which corresponds to part of a parabola. 
The cannon ball problem which Galileo tackled is identical with that in 
which the aeroplane is rising steadily as it drops the bag. If the ball leaves 
the gun with a velocity v at an angle a, it would move through a distance 
d = vt reaching P in t seconds, if gravity did not pull it earthward. It would 
then be h feet above ground, instead of y the height to which it has fallen 
meanwhile. The height through which it falls in t seconds is therefore h y, 
and 

The figure shows that 

h = d sin a 
= vt sin a 
,\ y = vt sin a \gt* 

The horizontal displacement along the x axis during the first t seconds is 
x vt cos a, so that t = x ~ v cos a. If we put this for t in the last expres- 
sion for jy, 

x sin a gx 2 

cos a 2v~ cos 2 a 

When the ball reaches the ground y = 0, so that 

gx = j 

2z> 2 sin a cos a 

Since sin 2a 2 sin a cos a: 

v* sin 2 a 



This gives the range of the cannon ball. To get the maximum height, sub- 
stitute half this value of x in the original formula for y. 

In applying the law of terrestrial gravitation to the range of the cannon 
ball, Galileo was the first to use the principle of inertia. The obvious limita- 
tions of this law did not escape his attention and his confidence in applying 
it was based on simple considerations. Since wood shavings fall slowly, while 
a wooden ball falls from the same height in approximately the same time as 
a cannon ball, we can conclude that air resistance does not make much 
diiference to the movement of compact solid bodies. Newton showed that a 
coin and a feather keep pace in a long cylinder from which all the air has been 
pumped out, when the cylinder is turned upside down. He thus concluded 
that the presence of air is the only circumstance which limits the rule that all 
bodies fall to earth with the same acceleration. That air itself is pulled earth- 
ward is shown by the very existence of the atmosphere, and we see later 
(Chapter VIII) that the behaviour of balloons is no exception to the more 
general law that all matter falls earthward with the same acceleration at the 
same place in a vacuum. We now know that air friction increases enormously 
at very high speeds. So it cannot be neglected in the practice of modern 
artillery. Calculations which gave a tolerable precision when applied to the 



268 



Science for the Citizen 



slow-moving short-range projectiles of Galileo's time would lead to grossly 
inaccurate conclusions about the shells used in warfare today. When Big 
Bertha shelled Paris at a distance of 76 miles with shells rising to a height of 
24 miles and a muzzle velocity of 1 mile a second, its actual range and the 
height of projection were just less than half the figures calculated by the 
method shown in Fig. 171. 

Apart from the big resistance of the air when modern projectiles are fired 
at very high speeds, the method Galileo used would not be correct for another 



80 



If the, proj(2cHk were, nof 
pulfad d&vm by gravity it 
would reach, iivis pomt in. 
t seconds 




'Position after t seconds 
x = d cos 55% 



\ 







\ 



\ 



displacement 
d cos 55% 

along ihe 
x axis 



60 



740 



160 



O 20 4O 

*-~x d cos 55^1 -> 

FIG. 172. AIR FRICTION AND RANGE OF PROJECTILE 

Neglecting air resistance and gravity the projectile would reach Q in t seconds travelling 
with velocity v at 55J to the ground. 

Neglecting air resistance only the distance travelled before hitting the ground is 
given by 

v z sin 2a , 0/ ,_ x 

x = ( see p> 267) 

o 

This calculation would be strictly true in a vacuum, where there is no air resistance. 
In real life there is considerable friction in the passage of the projectile through the 
air. Experiment shows that the frictional resistance of the air increases with the 
velocity of projection^ and is small at low speeds. Hence the formula gives a fairly 

food rough estimate for a slow cannon ball like those used in the time of Galileo, 
t is grossly inaccurate for modern artillery which can project shells like those of Big 
Bertha at 1 mile a second. Big Bertha shelled Paris 76 miles away at an angle of 55J. 
Neglecting air resistance the shells would have fallen at twice this distance, i.e. (since 
1 mile = 5,280 feet). 
(5,280) 2 sin 111 



32 



-feet = 5,280 x 165 x sin 69 feet = 165 X 0-934 miles = 154 miles. 



Its maximal height of 24 miles was less than half the height 56 miles given by the 
graph of x and y. 

reason. Over a short range we can regard the earth as flat, the direction of 
the plumb line as everywhere parallel, and the value of g as fixed. We shall 
see later that g falls off quite appreciably at a high altitude, and the angle 
of the plumb line rotates through a degree in a 70-mile range. So it would be 
more correct to reckon the acceleration of the shell at each stage of a long 
course as if it were directed towards the centre of the earth, and we should 
have to make allowance for the fact that its numerical value as well as its 
direction changes during the shell's journey. 



Wheel, Weight, and Watchspring 269 

When anything moves in a curved path, like a cannon ball, we can often 
look on its motion as if it were made up of a fixed velocity along one line 
and an acceleration directed to some fixed point. If it moves with a fixed 
speed in a complete circle, it has an acceleration towards the centre. When 
we make a stone swing in a circle at the end of a string, the latter exerts a 
pull on the fingers, or, to put it in another way, the fingers exert a pull on the 
stone to keep it from flying off at a tangent. Since pulling power implies 
acceleration, there is a continuous acceleration of the stone to the centre. 
At first sight this may seem to be a paradox because the stone keeps the 
same distance from the centre throughout its journey. The paradox dis- 
appears when you recollect what it would do, if the string were cut. It would 
fly off at a tangent. The very fact that it keeps a fixed distance from the centre 
implies that at each stage in its journey it is falling in towards the centre 
from the path it would follow if no string were pulling on it. Like the long- 
range projectile fired horizontally from a high mountain it has two motions, its 
own inertia which drives it along the tangent to the circle, and an acceleration 
towards a centre. When its implications were fully understood, this novel 
aspect of motion in a circle revolutionized the technology of clock-making, 
and the theory of planetary orbits. 

CENTRIFUGAL MOTION 

Before any substantial progress could be made in the theoretical design 
of seaworthy clocks it was necessary to understand rotary motion. The 
problem of range-finding is to reconstruct the curved path of the cannon 
ball from the circumstances of its motion. Rotary motion sets us the converse 
problem of reconstructing the central acceleration of a moving body whose 
path is known. If we could make a film of the path of a mail bag dropped 
from an aeroplane flying at a known horizontal speed we could calculate g 
from the graph. When we rotate a stone in a circle at fixed speed we know the 
path, and the calculation is comparatively simple. 

The chief thing which is likely to cause difficulty is that the word accelera- 
tion is now used in everyday speech for increasing (or decreasing) the speedo- 
meter reading without regard to the course itself. As long as the course is a 
straight one the car driver's acceleration is the same as Galileo's. This is not so 
when the car is turning a bend. If Galileo's acceleration merely meant change 
in the reading of a speedometer, no pull would be needed to change the direc- 
tion of a car which kept a fixed speedometer reading while turning a bend. 
Acceleration as a measure of pulling power is not necessarily the same as 
change in speedometer reading. It is based on stop-watch readings like those 
of speed-cops watching the car as it passes two fixed points at the ends of a 
surveyor's chain. This comes to the same thing if the car's course is a straight 
one parallel to the chain. 

When we have to deal with motion in a curved path, the ordinary meanings 
of the words speed, velocity, and acceleration, as they are used loosely in 
everyday life, have to be modified to take account of the principle of inertia. 
A weight can be applied to affect the motion of an object in either (or both) 
of two ways. One is to change its speed along its own straight path. When this 
is so, the velocity of the moving object is numerically the same thing as its 



2jo Science for the Citizen 

speed, and the acceleration imparted to it merely signifies increase or decrease 
of speed in one direction or its opposite, as in the everyday use of the words. 
A weight can also deflect the straight line motion of an object from the direc- 
tion in which it is moving. So, if pulling power depends on the acceleration it 
can impart, acceleration must be measured so that it implies change of direction 
as well as change of speed. This can be done without sacrificing its ordinary 
meaning as applied to motion in a straight line, like motion down a flat slope 
or vertical motion under gravity. To make it measure change in direction as 
well as change in speed (actual distance traversed in one second), is not 
difficult if we recall how we measure direction when we have to be precise 
about it, 

To measure the direction in which a body is moving we have to select 
some fixed line or plane of reference. Commonly we choose the N-S meridian 
or the line joining the east and west points on the horizon, when we are 
speaking of terrestrial motion like that of a ship. We use the meridian and 
the plumb line when we are talking of celestial motions like that of the 
sun. We measure the direction of a movement by the angle which the fixed 
line or plane of reference makes with the path of the moving body at any 
instant if its path is straight, or with the tangent to it if it is turning a 
corner. The distance which a moving object traverses in a given time depends 
on the direction from which we are looking at it. If we call its progress measured 
along a line at right angles to the direction in which a person is looking 
at it, its motion relative to the observer, its movement relative to observers 
looking at it from different aspects will generally be different. In what follows, 
we shall assume that the observer is a long way off. The "lines of sight" are 
then parallel for the same person. 

The observer who watches the progress of the car in Fig. 173 by the 
telegraph poles along the road AC, which is its actual path of motion, sees it 
pass five gaps. During the same time an observer watching it pass the tele- 
graph poles on the road BC sees it pass only three of the gaps between the 
poles at the side of the latter. A third observer counting the telegraph poles 
along the road AB as it passes them, sees it pass four gaps. Relative to the 
three observers the distances covered during the same time are in the ratio 
5 : 3 : 4. If one gap is the unit of distance, the first of these is the speedo- 
meter reading, i.e. its speed in the ordinary sense, which signifies the actual 
distance traversed by the car in unit time. Speed, then, is distance covered in 
unit time relative to an observer looking along the line at right angles to the 
actual path. If the path itself is curved, the observer can only do so by 
changing his point of view. So speed, as the term is ordinarily used, does not 
imply motion relative to a fixed observer, i.e. to an observer watching it with 
reference to the same fixed line. 

In everyday speech we draw no distinction between speed and velocity. If 
we limit the use of the word velocity to the measurement of motion relative to a 
fixed observer, there is no difficulty in adapting our definition of acceleration 
as increase (positive sign) or decrease (negative sign) of velocity in unit time 
to the measurement of a pull when it only affects the direction without changing 
the speed with which an object is moving. As long as a body moves in a 
straight line the actual speed or distance traversed in unit time is numerically 



Wheel, Weight, and Watchspring 271 

the same as its velocity relative to an observer looking along a line at right 
angles to its path, i.e. its displacement in unit time along the line of reference. 
Its acceleration is therefore numerically the same as the rate at which its 
speed is changing. On the other hand, an object moving with a fixed speed 
round a corner cannot move through equal distances, measured parallel to 
the same straight line, in equivalent intervals of time. So it cannot have 



To this observer the car 

has reached a, point 

halfwag between, t-he 

y third & fourth, polos' 

ontiier&ad AC 




To this one itr" 
reached lia}f- 
wag bdwzeti the, 
second & third, 
pole* aimg 



one eves- it 

Ac third pole 
along AB 

FIG. 173. MOTION RELATIVE TO DIFFERENT FIXED OBSERVERS 

For reasons stated in the text, the observers stand at a distance much greater than 
can be shown in a diagram. 

a constant velocity relative to any fixed observer. A car (Fig. 174) which 
passes the milestones on a circular by-pass in equal intervals of time would 
give a constant speedometer reading, while its progress relative to the 
milestones on the main road would exhibit a continual decrease. So if a 
body moves in a curve at fixed speed, or in other words if it is changing 
its direction without changing its speed, it will have an acceleration relative 
to any fixed observer. The driver of the car in Fig. 174 might regulate his 
so that he always appeared to pass the milestones along the rnain 



272 



Science for the Citizen 



road in the same interval of time. Although he would then have no accelera- 
tion with reference to the main road, he would still have an acceleration 
(cf. Fig. 175) with reference to a road at right angles to it. Thus change of 
speed or change of direction always involves acceleration of motion (positive 
or negative) relative to some fixed observer. 

At every point in its course a weight moving at fixed speed in a circle 




LL~.+t. ti f ^yjiHt&. .-fS^nvJ 5&V^ 

- */ ' ' ' .- * vc r 3 




FIG. 174. CONSTANT SPEED IMPLIES ACCELERATION 

The car passes the milestones A, B, C, etc., on the by-pass in equal intervals of time. 
Hence its speedometer reading does not change. Its motion relative to any fixed straight 
line is continually changing. For instance, relative to the main road it passes through 
the distance ab while it moves on its own path through AB, and while it takes the same 
time to move from F to G it only traverses fg relative to the main road. Thus its 
velocity decreases as it appears to move away frcm the cross road which forms the 
centre of the arc. If it moved in the opposite direction, its velocity would appear to 
increase. It has a negative acceleration along the diameter to its path as it moves away 
from the centre. 

has an acceleration along the line joining it to the centre. This acceleration 
measures the rate at which it must be pulled in to counteract its own sluggish- 
ness which would carry it forwards along the tangent. If it makes one revolu- 
tion in a circle of radius r feet during T seconds (called its periodic time), 
it actually moves through 2?rr feet with a speedometer reading of (2?7r) -f- T 
feet per second. Having reached some stage represented by P in Fig. 176., 
it would continue to move in the straight line along the tangent to its circular 
course if there were no force pulling it to the centre. At P it is actually moving 



Wheel, Weight, and Watchspring 273 



/ 



Distance traversed relative, to 
X-axis 

FIG. 175. CHANGING DIRECTION ALWAYS IMPLIES ACCELERATION RELATIVE TO 

SOME OBSERVER 

The figure shows successive positions' at the end of each second during a 5-secpnd 
movement of an object in a curved path. Its speed as well as its direction is changing, 
being obviously greater in the last than in the middle second interval. Relative to the 
X-axis the change in speed makes up for the change in direction, and the distance 
traversed in unit time is the same and is always to the right. Relative to the Y-axis 
its velocity changes proportionately more than its speed along its actual path. 




FIG. 176. WEIGHT REVOLVING IN A CIRCLE AT FIXED SPEED s FEET PER SECOND AT 
THE END OF A CORD r FEET LONG 

If there were no pull exerted by the cord, i.e. if the cord were cut at the instant when 
the weight reaches P, it would fly off at a tangent, reaching Q during the time in which 
it would otherwise get to R. At the instant when it is at P its direction of movement 
is along PQ. Hence its speed s is numerically the velocity with which it would con- 
tinue to move of its own inertia through the distance c = PQ in time r, i.e. c = st. To 
bring it to R during the same time, we have to supplement its motion in virtue of its own 
inertia, like that of the mail bag in Fig. 170 with a motion parallel with RO at each 
stage in its course. This must make it fall a distance b along RP while it would other- 
wise reach Q If its acceleration along RO is a, b iar*. 



274 Science for the Citizen 

in this direction. So its velocity along the tangent would also be (27rr) -f- T 
if the cord were cut when the weight reaches P. In a short interval of 
time t seconds it would then traverse a certain distance c feet reaching Q. 
Actually it does not reach Q. The force exerted by the string keeps it moving 
inwards, so that it is at a fixed distance r from the centre. Just as the mail bag 
continues to move with the horizontal velocity of the aeroplane, while falling 
simultaneously with an acceleration g earthward, the weight would travel 
from P to R if: 

(a) it continued to move forwards along PQ with a velocity 5 (= 2nr -:- T) 
through a distance c in time / (see Fig. J 76), 

(b) it also fell inwards during the same time t along a direction parallel to 
RG through a distance b with fixed acceleration a. 

The definition of velocity tells us thot s ~~ c - t, or c = st = 2-n-rt ~ T. 
The rule connecting acceleration xnd distance tells us that b \at l . The 
theorem of Pythagoras shows us the connexion between the distances b and 
c, i.e. 

(r -f b)* -- r' + r 2 
/. 2br + b 2 - c 2 
:. 2b(r + Afr) -= c 2 
.*. at*(r + IV) -- 4?rW -f- T 2 



When the weight has not moved appreciably beyond P the distance b does not 
differ appreciably from zero, and the direction of QO does not differ appre- 
ciably from PO. So, if we neglect &, a becomes the instantaneous acceleration 
along the radius when the weight is at P, and 



If the mass of the weight is m the force it exerts on the string is therefore 

F --- m . a 



T' - 

If the circular speed is n revolutions per second it takes I ^~ n seconds to 
rotate once and 



Hence we can also write for the force exerted on a mass of m rotating at a 
Distance r from the centre of a wheel making n revolutions per second 



Wheel, Weight, and Watchspring 275 

If m is measured in pounds, n in revolutions per second, and r in feet, this 
gives the force in poundals. If m is measured in grams, n in revolutions per 
second, and r in centimetres, it gives the force in dynes. 

There are two ways of representing the motion of an object revolving at 
/ixed speed in a circle. Each is useful, and the two are complementary. In 
a tug of war, when each side is keeping the same distance from the line, the 
two teams are pulling with equal strength in opposite directions. The forces 
exerted are numerically equal, and we represent them with opposite signs 
to signify that their sum in either direction is zero. Hence there is no motion. 
In the preceding paragraphs we started with the principle of inertia, that is 
to say with the known fact that the weight would fly off at a tangert if the 
cord were cut. We then asked what acceleration towards the centre is neces- 
sary to keep the weight at the same distance from it. We are equally entitled 
to put the question in the converse form. If a weight revolving in this way 
has an acceleration towards the centre, we must also say that it has an equal 
acceleration away from the centre to keep it at the same distance from it. 
The first or centripetal acceleration (27rw) 2 r depends on the cohesive power 
of the cord which resists any attempt to stretch it. The second or centrifugal 
acceleration + (27rw) 2 r is a mathematical trick put in to signify that there is 
no actual motion towards the centre. Because of the power of the cord or 
support to resist stretching, a revolving weight is pulled towards the centre 
of motion. As in a tug of war, the two pulls, away from and towards the 
centre, are balanced. If we use the positive sign for the outward direction, 
it comes to the same thing whether we say that the centripetal force exerted 
on the weight by the cord is (2Trri) 2 mr or that the centrifugal force exerted 
by the weight on the cord is + (Zrrrifmr. 

The practical importance of the distinction is easier to see, if a piece of 
elastic is used instead of an ordinary cord which does not appear to stretch 
appreciably. It is a matter of common experience that a large weight stretches 
a piece of elastic more than a small one. When a weight hangs vertically from 
a piece of elastic, the power of the latter to resist further stretching cancels out 
the action of gravity and must therefore be represented as a force of equal 
strength and opposite sign. The resisting force of a piece of elastic therefore 
increases with the load. It is easy to satisfy yourself that a weight swung in 
a circle at the end of a piece of elastic stretches the elastic more when the 
speed is increased. As we increase the speed, the centrifugal force outwards 
from the centre is also increased. The weight moves farther away from 
the centre, thereby increasing its centrifugal force, while at the same time 
increasing the power of the elastic to resist any further stretching. When 
the latter just balances the former, the weight continues to revolve at a 
fixed distance from the centre. If the speed is reduced, the centrifugal force 
is weakened. The elastic resistance being now the greater draws the weight 
inwards. Since diminishing the length diminishes the power of the elastic 
to resist stretching (see p. 291), there comes a point when the elastic pull is 
no longer in excess, and the weight continues to revolve at fixed distance from 
the centre, though nearer to it than it was when the speed was greater. 

To test the law of motion in a circle we have only to attach a weight to a 
spring balance fixed by a ring to the axle of a turntable, on which the weight 



276 



Science for the Citizen 



can slide without appreciable friction (Fig. 177). When the table is made to 
rotate the weight will move outwards and stop at a certain distance from the 
centre. This distance r depends on the speed. The spring balance records 
the pull of a weight hanging vertically. So an extension marked M lb corre- 



Oulwa.rd pull of 




FIG. 177. MEASURING CENTRIFUGAL FORCE 

sponds to 32M poundals. If the mass m is rotating at n revolutions per second 
r feet from the centre, 

32M - (27rn) 2 mr 

If the mass attached to the spring balance is 2 Ib., and it remains steady 
2 feet from the centre when the wheel rotates at 20 revolutions per second, 

/44 \ 2 

32M- (- X 20 X 2 x 2 



Taking 4rr 2 as approximately 40 

M = 2,000 Ib. 



Wheel, Weight^ and Watchspring 277 

That is to say, the scale reading of the spring balance would correspond to a 
hanging weight of 2,000 Ib. 
The construction of a variety of devices based on this principle provides 



x = 



y-ingr 




Ike steapkanacr tukctvtke train, 

^ P starfar 



Backward swun of 
legs 
fo 





of free end! 
valaifc to aa 
oufward puZL 




FIG. 178. PRINCIPLE OF THE CREAM SEPARATOR OR CENTRIFUGE 

To calculate the angle of tilt of the two tubes rotating on a bar of radius r at a given 
speed 

2 47TV 

r -. mg = -^ 

Thus if the distance of the fixed end from the centre is 1 foot and the axle makes 
6,000 revolutions per minute the time of a revolution is 60 ~ 6 3 000 = 0-01 second. 
Taking 4V 2 = 40 and g = 32 feet per sec 2 . 



tan a == 



40 x 1 



= 12,500 



32 X 0-01" 

Since tan 89 = 57-29, the tubes are inclined at over 89 to the vertical, and would 
hence be practically horizontal at this speed. 

tangible evidence of this pull. Examples from the everyday life of our own 
time are the cream separator, the governor of a steam engine, and the analo- 
gous device with air vanes used as a brake to ensure the constant speed of a 
gramophone record. The cream separator is essentially like the laboratory 



278 



Science for the Citizen 



centrifuge, which consists of a rotating bar or flywheel, from the rim of which 
a series of tubes are suspended so that the lower end is free to tilt upwards 
and outwards (Fig. 178). During rotation the centrifugal acceleration 
(27ra) 2 r at the top of the tube is balanced by the rigidity of the support, while 
the opposite end is pulled downwards by gravity and the resultant pull R 
(Fig. 178) makes it tilt outwards during rotation. At high speed the pull 
outwards is far stronger than gravity, and makes cream rise or sediment 
settle much more rapidly than it would do if left to the action of gravity 
alone. In the governor of a steam engine (Fig. 179)., the two metal balls are 




FIG. 179 

How the Governor of a steam engine allows the escape of steam when the speed exceeds 
the danger limit depends on the same principle as the centrifuge. Two metal balls are 
each connected by a pair of levers with two collars A or A' and B. The weight of 
the balls keeps them in the position shown on the left when the engine is at rest. As 
the shaft revolves the balls fly outwards as shown on the right lifting up A into position 
shown as A'. A lever attached to A is connected to a throttle- valve in the steam pipe 
leading to the cylinder which closes when the collar A' rises to a certain height 
depending on the speed of rotation. Only one ball is shown in each position. 

connected with a lever device. At a certain speed the balls fly apart so far 
that the lever closes a throttle controlling the supply of steam to the cylinder, 
so the speed cannot exceed the limits of safety. A common type of speed 
regulator for gramophones has a lever attachment like that of the steam 
engine governor, working a brake when the speed reaches the limit required. 
In the analogous device which is used to regulate gramophone speed, 
the air-resisting surface of the vanes increases and thereby opposes further 
increase of speed. 

The calculation of centrifugal force is also necessary for laying rails at 
appropriate heights to allow for the tilt of trains in turning a bend. In its 
own social context it arose from the technology of the pendulum clock, and 
the formula was first actually published in Huyghens' Horologium Oscilla- 
torium. The first clocks were purely empirical devices, in which a descending 
weight was the motive power. Later the weight was replaced by a spring. 



Wheel) Weight^ and Watchspring 279 

The problem of clock construction is to regulate the fall of a weight, or the 
contraction of a stretched spring, so as to drive a wheel at a constant speed. 
This is done by a device called the escapement (Fig. 181), which consists of 
two arms cut so that one is free when the other engages the obliquely toothed 
rim of the driving wheel, and vice versa. As the driving wheel revolves, the 
pressure of a tooth swings one arm up. As it clears the tooth, the other comes 
down between a pair of teeth, engaging them till its fellow swings down. 
The motive force is thus regulated to impart constant speed by the regular 
swinging of the escapement to and fro. So accuracy of timekeeping depends 
on making the interval of the escapement swing constant. 

In the first clocks this was done by what was called the "verge escape- 
ment" (Fig. 181). The escapement was fixed to a bar with a heavy weight at 
each end. First it was turned one way by one tooth of the driving wheel. 
Then another tooth on the opposite side swung it the other way. If the 
teeth of the rim were of the same size and cut at the same angle, the bar 



\ 
\ 
\ 
\ 

\ 



tan a = 




FIG. 180 

Inclination (a) of rider on a circular speed track as the resultant of two accelerations 

at right angles. 

and weights turned through the same angle in opposite directions taking 
the same interval of time for each swing. To make this arrangement work 
with great accuracy therefore required very great precision in the construc- 
tion of the wheel and the escapement; and since the heaviness of the weights 
generated considerable friction, calculation of the dimensions was beset with 
formidable difficulties. Accuracy in clock construction was not achieved until 
it was possible to devise an escapement which has a fixed period, independent 
of the size of the teeth on the driving wheel. This can be done by means of 
a pendulum or a spring. 

THE PRINCIPLE OF THE FIXED PERIOD 

The rule for rotational motion leads directly to another rule about the 
circumstances of to-and-fro motion with a fixed period. In the steam engine 
the to-and-fro stroke of the piston is converted into the rotary motion of 
the wheel. Conversely if we rotate the wheel when there is no steam in the 
piston we can convert the rotary into a periodic motion. It is not difficult to 
see that a wheel rotating at fixed speed can be made to produce a pendulum- 
like movement with a fixed periodic time. The complete period, i.e. the time 



280 



Science for the Citizen 



taken to move forwards and backwards to the same position, is the time 
taken for the wheel to make a complete revolution, i.e. its periodic time T. 
Fig. 182 shows a type of piston rod attachment by which the horizontal 



Isczponzrt 




(2?) Anchor Eszap 





FIG, 181. EVOLUTION OF THE PENDULUM CLOCK 

A pendulum swinging in a small arc of a circle is almost but not quite isochronous (i.e. 
with a period independent of the distance through which it swings). It is completely so, 
if the path of the bob is a cycloid, which is a curve like a semi-circle but steeper at the 
extremities. By having a flexible support which bends round the curved jaws shown 
in the figure, the pendulum performs a cycloidal motion for wide angles of swing. 

displacement of a peg projecting from a circular wheel drives a pendulum 
to and fro. The horizontal displacement of the centre of the pendulum bob 
from its midway position exactly corresponds to the horizontal displacement 
of the pin along the diameter of the wheel, and hence the acceleration of the 



Wheel, Weight, and Watchspring 281 

pendulum bob is also equivalent to the acceleration of the peg along the 
diameter of the wheel. 

Suppose now that at some instant during the rotation of the wheel in 
Fig. 182 the radius r drawn from the centre to the peg makes an angle b 
with the axis of the piston. The peg is moving in a circle at fixed speed. 
Inwards along r (see Fig. 176) it has an acceleration whose numerical value 
may be represented by a r . According to rule 4, p. 258, this is equivalent to 
an acceleration a r cos b along the piston axis (see also Fig. 174). If the 
numerical value of this horizontal acceleration is a X9 

a r a r cos b 




FIG. 182. REGULAR PENDULUM-LIKE MOTION OF A PIN ATTACHED TO THE RIM OF A 
WHEEL ROTATING AT A FIXED SPEED, AS SEEN IN THE HORIZONTAL PLANE 

If x is the horizontal distance of the peg from the centre (regardless of 
whether it is right or left of it) cos b = x ~ r, 

:. a x ^a^~r 

.'. a x -r- x =-- a/-:- r 
And since a r and r are both fixed 

~ = const 
x 

The pendulum of Fig. 182 swings with a fixed period corresponding to 
one rotation of the wheel at fixed speed. We now know that when it does so, 
the ratio of the numerical values of its acceleration and displacement along 
the line of swing is fixed. This clue is not enough to define a periodic motion 
of fixed period. As it stands, the last formula takes no account of the direction 
of motion, and would also describe that of an object moving continuously 
with increasing speed in the same straight line. The motion of the pendulum 



282 Science for the Citizen 

involves reversal of direction in each revolution with slowing down and 
speeding up in each half turn. To define its direction we must use appropriate 
signs. 

If x is the horizontal distance of the peg from the centre, its displacement 
may be. x (x ft. to the left) or + x (x ft. to the right of it). With the same 
convention an acceleration + a x means gaining a x ft. per sec. in each second 
while moving to the right or losing a x ft. per sec. 2 while moving leftwards^ 
and a x means gaining a x ft. per sec. 2 moving leftwards or losing a x ft. per 
sec. 2 moving rightwards. In a complete clockwise rotation the horizontal 
displacement is + x, when the peg is moving downwards, and x, when 
the peg is moving upwards. In the downward half turn (see Fig. 174) its 
acceleration is a x) because it is first slowing down as it moves right, and 
then speeding up as it moves left. Conversely, it is + a x in the upward 
half turn, when it is first slowing down rightwards and then speeding up 
leftwards. With the usual conventions of sign the horizontal acceleration is 
therefore a x when the displacement is + #, and + a x when the displace- 
ment is x. To take account of this we replace the constant in the previous 
formula by a positive quantity k to which the negative sign is fixed. If we 
write, 

a x = kx 

a x now stands for the actual acceleration with its appropriate sign., and x for 
the actual displacement also with its appropriate sign. Thus if the displace- 
ment is d ft. to the left, x = d and a x = k ( d) = + kd, which is 
positive. If the displacement is d ft. to the right, a x kd, and since k 
and d are both positive the acceleration is represented by a number to which 
the negative sign is attached. This signifies slowing down in motion to the 
right or speeding up leftwards. 

Since a r -~ r = a r ~ x, a r -f- r - &, and we know that the numerical 
value of a r , the centripetal acceleration, (p. 275) is (2?! -f- T) 2 r. 

/. a r -r- r = (277 -r T) 2 

/. T - 277 + Vk 

So we have now two clues to the perfect clock escapement. We have to find 
something which moves so that it has an acceleration of opposite sign and 
with a constant ratio (K) to its displacement along a fixed line. When we 
have done so its periodic time will be 2?! ~ Vk. 

To visualize a periodic motion with correct use of signs, complete the 
following table showing the mean horizontal acceleration of the pendulum 
device in Fig. 182 in successive seconds. The wheel rotates once in 36 seconds, 
i.e. 10 per second. Starting when the pin is at the extreme right-hand position, 
its horizontal displacement x is also r, the distance of the pin from the centre, 
so if r = 1, x = 1. At the end of a second the pin has rotated through 10 
and x -= r cos 10 = cos 10 At the end of two seconds x = cos 20 and so 
on. Tables of cosines tell us that cos 10 = 0-9848 and cos 20 = 0-9397. 
During the first second the displacement changes from x = 1 to x = 0-9848, 
i.e. 0-0152 ft. to the right. In the next second it moves 0-0461 ft. Thus 



Wheel, Weight^ and Watchspring 283 

its mean velocity in the first second is 0-01 52 ft. per sec. and its mean velocity 
in the next second is 0-0451 ft. per sec. and the mean velocity gained in 
one second (mean acceleration) is 0-0451 ( 0-0152)= 0-0299 ft. 
per sec. 2 This is its approximate acceleration midway between the middle of 
the first and the middle of the next second, i.e., at the end of the first when 
its distance is 0-9848 ft. The ratio of its acceleration to its distance is 0-0299 
~ 0-9848 0-0304. The table below calculated in this way shows that 
the ratio a x -f- x is approximately constant, and closely agrees with the 
value of 47T 2 ~ T 2 = 4 X 9-87 + (36) 2 = 0-0305. 



Time 


Angle 

(o) 


Displacement = x 


Mean Velocity 
(Displacement 
gained per sec.) 


Mean Acceleration 
(Velocity gained 
per sec.) = a r 


(If ' X 








1-0000 








0-5 






0-0152 






1-0 


10 


0-9848 




- 0-0299 


0304 


1-5 






- 0-045] 






2-0 


20 


0-9397 




- 0-0286 


0-0304 


2-5 






- 0-0737 






3-0 


30 


0-8660 




0-0263 


0-0304 


3-5 






- 0-1000 






4-0 


40 


0-7660 




- 0-0232 


0-0303 


4-5 






- 0-1232 






5-0 


50 


0-6428 









THE LAW OF THE PENDULUM 

So long as the escapement was unreliable, it was neither possible to make 
a dock with sufficient accuracy for finding longitude at sea nor to fix a con- 
venient unit for measuring short intervals of time. Galileo seems to have 
been the first person who recognized that a pendulum device might be 
used to make an escapement whose period does not depend on the angle 
of swing, and is therefore independent of the precision of the teeth on the 
wheel. The principle of the pendulum is that when a small weight swings 
through a small angle, at a fixed distance from a fixed point, its time of swing 
is constant. It is said that Galileo noticed this phenomenon, as a student, 
when he timed the swing of a lamp suspended from the roof of a church 
by counting his pulse beats. It is more probable that the practice already 
existed among physicians, and that when he subsequently made accurate 
clinical observations on pulse rate by using a pendulum to count the heart 
beats of patients in high fever, he merely adopted the practice of contem- 
porary medicine. Be that as it may, his important contribution was to show 
that the period of the swing is approximately constant by mechanical 
standards, which had been used from time immemorial to measure astro- 
nomical time. He determined the number of swings of a pendulum during 
the time occupied by the emptying of a vessel; found that it did not vary 
appreciably for larger and smaller swings; suggested its use in observatories 
for measuring short intervals between the transits of neighbouring stars; 
and designed, without completing, a clock with a pendulum escapement. 



284 Science for the Citizen 

Having discovered the first simple and reliable means of measuring short 
intervals of time, the next problem was how to construct a pendulum of 
which the swing occupies some convenient fraction of a standard long interval 
of time, e.g. a sixtieth of a minute, or, as we now call this new unit, a second. 
By counting the number of swings corresponding to the flow of a fixed 
quantity of water from a vessel, he established the following conclusions : 
(a) within wide limits the time occupied by a complete swing ("period" 
of the pendulum) is not affected by the size or material of the weight, pro- 
vided that the length of the cord or rod is the same; (&) if the length of the 
pendulum is varied, the square of the period (T) is directly proportional to 
the length, i.e. 

T 2 = KL. 

To see what this means, suppose a bob hanging at the end of a cord 3 feet 
3 inches long makes 30 complete swings (i.e. from one position back to the 
same position) in a minute. The period is then 2 seconds, i.e. 

2 2 == K x 3J 
/. K- 16/13 = 1-23 

So to make a pendulum which gives a half swing in 2 seconds (i.e. one with 
a period of 4 seconds), the length is given by the equation 

4 2 = M x L 
.'. L = 13 feet. 

In a clock the escapement engages the teeth every half period. So what 
is called a seconds pendulum is a pendulum whose half period is one second 
(i.e. T = 2 sees.). Shortly after Galileo's death his French pupil Mersenne 
(1644) determined the length of a seconds pendulum with great care, and a 
little later Huyghens made the first successful pendulum clock (Fig. 183). He 
also modified the form of the escapement ordinarily used so that the motion 
would withstand the rocking of a ship, and at last it seemed as if Huyghens' 
marine clock had achieved what his fellow countryman, Gemma Frisius, had 
dreamed of more than a century since. 

This was not to be. A pendulum clock loses time, if taken from Paris to 
Cayenne in French Guiana. This is because the swing of the pendulum 
depends on the pull of gravity, and the latter is not exactly the same in all 
parts of the world. It weakens as we get away from the earth's centre in 
climbing a mountain (p. 300). It is also different at sea-level in different 
latitudes, and this would still be so if the earth were perfectly spherical. 
The latitude variation provided a new means of testing the Copernican 
doctrine. Long before Copernicus adopted it as part of his system for 
calculating planetary motions, the Pythagorean brotherhoods, the Athenian 
Archytas, and Aristarchus of Samos, had in turn toyed with the earth's axial 
motion as a speculative possibility. In Kepler's time it still seemed to contra- 
dict everyday experience, and was justified only by arithmetical convenience. 
The practical failure of the first marine clock brought it to earth. What was 
a serious setback to practical achievement set the stage for the new theories 
associated with Newton's name. 



Wheel, Weight, and Watchspring 285 

Discovery was now becoming an organized social institution. One of the 
cultural offshoots of "Colbertism" in France was the foundation of the Paris 
Academy of Sciences in 1666, four years after the Royal Society received 
its charter in England. As the latter grew out of the meetings of the Invisible 
College (see p. 552), the Paris Academy arose from an association of a group 
which used to meet at the cell of Mersenne. Mersenne was active in spreading 
Galileo's teaching. The original members included Descartes, Pascal, the 
mathematician Fermat, and Gassendi, whose commentaries on Epicurus 
revived the atomistic speculations of the early Greek materialists. In con- 




tecondt. 



FIG. 183. HUYGHENS' MARINE CLOCK 

formity with Colbert's policy the Paris Academy, like the English Royal 
Society, was actively interested in all problems related to navigation, then 
the touchstone of mercantile supremacy. Under its auspices the Paris 
Observatory was inaugurated and completed three years before the one at 
Greenwich was established as a national undertaking. A rich harvest of dis- 
:overies followed immediately. To Paris came Cassini from Italy and Romer 
from Denmark. Cassini undertook the calculation of tables forecasting 
2clipses of Jupiter's satellites for use in determining longitude at sea. The 
project was undertaken in accordance, Professor Wolf tells us, with a sug- 
gestion made by Galileo himself. A remarkable discovery to which Romer 
was led in the course of a similar enquiry will emerge in the next chapter. 
The Academy sponsored several expeditions, notably one to French Guiana 
with a view to simultaneous observations on the parallax of Mars from the 



286 



Science for the Citizen 



Paris Observatory and Cayenne (Lat. 5 N.)- This expedition, which gave 
the first relatively satisfactory scale of measurement for the planetary system, 
signalizes an epoch in clock technology. 

The technology of the clock, like other problems of navigation, was a 
prominent feature in the researches encouraged by the Paris Academy and 
undertaken by Mersenne himself. Twelve years after the invention of 
Huyghens' pendulum clock (1657), Picard, one of the foremost astronomers 
of the Academy, made careful measurements to determine the length of a 
sideral seconds pendulum by star observations at Paris and Lyons. At the 
request of the Academy, in the expedition of 1G72, Richer observed the 




FIG. 184. THE PENDULUM AND THE INCLINED PLANE 



length of the seconds pendulum at Cayenne. On returning to Paris he found 
that the same pendulum must there be lengthened by 1 Paris lines (12 to 
the inch). A year later Huyghens published the theory of the pendulum clock 
(Horologium Oscillatoriunf). In this he explained the retardation of the clock 
by the earth's axial motion, established the mechanical principle involved 
in rotation at constant speed, and arrived at the correct conclusion that the 
earth must be slightly flattened at the poles. 

We can see why a pendulum clock cannot be used as a reliable chrono- 
meter at sea by applying our two clues to the perfect clock (p. 282) and 
the law of the inclined plane. Turn first to Fig. 184, which illustrates a small 
weight swinging like the bob of a pendulum, in an arc of a circle of radius r. 
At any point in its course, we may consider it to be sliding down an inclined 
surface represented by the tangent to the curve at the point where it is, with 



Wheel, Weight, and Watchspring 287 

downward acceleration g sin b. From the geometry of the figure, sin b 
x -f- L; and its acceleration (a) to the right along the tangent is, therefore. 



Thus the acceleration is actually to the left. Along the line at right angles to 
the resting position of the axis of the pendulum, the equivalent acceleration a x 
of the bob is a cos b (see Fig. 1 64). Now cos b does not differ much from 1 
when b is small (e.g. cos 5 = 0-9962 differs by only 4 parts in a thousand 
from cos = 1-0000) at any part of its journey. For a 4-inch swing (2 inches 
each way) of a 36-inch pendulum the maximum angle is 3 2 from the vertical 
and cos 3 2 == 9984. So cos b differs from 1 by only 16 parts in 10,000 over 
the extreme limits of the swing. Even for a three times greater swing of 
12 inches or 6 inches each way when the maximum value of b is 9 5, cos b only 
varies between cosO 1-0000 and cos 9 -5 = 0-986, differing from the 
mean of 0-993 by 7 parts in a thousand. It also differs from the mean value 
0-9992 for a swing a third as great by less than 13 parts in a thousand. So a 
very small difference in the angle of swing due to irregularities of tooth cutting 
in a clock wheel can make only a negligible difference to the extreme values 
between which cos b varies. We may therefore draw this conclusion. To a 
high degree of precision for relatively small angles of swing the horizontal 
acceleration a x 

g , __ g 

= x cos o - _ x 

The horizontal acceleration towards the resting position is therefore pro- 
portional to x, the horizontal displacement, and the constant ratio k is g ~ L. 
The rule given on p. 282 shows : 

(a) that the pendulum swings with a constant periodic movement as we 
should expect if Galileo's principles are right; 

(b) that the constant ratio k must be (2?r ~ T) 2 so that 

T - 277 ~ Vk = 277 

This you can test easily for yourself by hanging a button on the end of a piece 
of thread and counting the swings with a watch. On measuring the length 
L of the thread you should get a value for g within 2 per cent of the best 
values with little trouble. You can also see that it gives the length of the 
seconds pendulum (period 2 seconds) given already. Taking (2?7) 2 as approxi- 
mately 40, L = g 4- 10 = 3-2 feet or 3 feet 2 inches. 

The important thing about the calculation we have just made is that the 
period of the pendulum depends on g. So if g varies at different places the 
pendulum clock can only be relied on to work properly while it is kept at the 
place where it was regulated. The Cayenne expedition showed that g does 
vary, and Huyghens found the answer. 

The earth like a great wheel makes a complete rotation about its axis with 



288 Science for the Citizen 

a periodic time (T) of 24 hours. A man who could work miracles might 
suspend the pull of gravitation as in the story by Mr. Wells. Any object 
would then fly off immediately at a tangent to its circle of latitude L (radius r). 
To keep an object from doing so, it would be necessary to impart an accelera- 
tion (2n -f- T) 2 r towards the earth's axis parallel to the plane of the equator, 
or (2n -f- T) 2 r cos L in the direction of the earth's centre. If the man 
who could work miracles decided to stop the earth's rotation instead of 
suspending the earth's gravitational pull, the effective acceleration which the 
latter could impart to a body would be greater. Objects would fall to earth a 
little faster, because gravitation would no longer be opposed by "centrifugal 
force." Although we cannot stop the earth's rotation we can get to the poles, 
where r is zero, and the value of g is not affected by the earth's spin. At any 
other latitude the pull of gravity has to do two things : (a) keep a body from 
flying off at a tangent to the earth's surface, (b) keep it moving towards the 
earth's centre if free to do so. When we measure its acceleration towards the 
latter, we are therefore measuring the difference between its acceleration at 
the poles where the pull of gravitation has only to do (b) and the acceleration 
which would keep it at a fixed distance from the earth's centre, if it kept to a 
circular path. The observed value of g at latitude L should therefore be: 

27T\ 2 

} r cos L 
and if R is the radius of the equator (see Fig. ] 85), this is 



At the equator L = and cos L = 1, R is 4,000 miles or 4,000 x 5,280 feet, 
and the time of a revolution is 24 hours or 24 x 3,600 seconds. Taking 4?r 2 as 
40, the observed acceleration is diminished by: 

4,000 X 40 X 5,280 

= 0-114 foot per second per second 



24 x 3,600 X 24 x 3,600 

The earth's spin therefore should reduce the value of g by 1-4 inches per 
second per second at the equator. How to make a corresponding calculation 
for any other latitude is shown in Fig. 185. The value of g thus diminishes at 
sea-level from the poles to the equator, and since the pendulum period varies 
inversely with g, T increases. That is to say, there are fewer swings in the 
same period of astronomical time, and the clock will be "slower" if taken to 
a latitude nearer the equator. A "marine" pendulum clock would only work 
during a course along the same parallel of latitude. It would also be useless 
for exploration, because the value of g is measurably smaller at the top of a 
high mountain and greater at the bottom of a deep mine than it is at sea-level. 
The explanation given refers to a simple pendulum, i.e. small weight 
suspended from a light axis such as a thread, at sea-level (see p. 300). We 
can then regard all the effective movement as the movement of the weight. 
To calculate the movement of an ordinary pendulum is a little more compli- 
cated, because we have to average out along the axis the effective weight 



Wheel, Weight, and Watchspring 289 

involved in the descent by the methods explained on pp. 610-611 in 
Chapter XII. Huyghens calculated the motion of the compound, i.e. ordi- 
nary, pendulum in his great treatise on the theory of dock motion, and 
also made another important discovery. There is as we have seen a small 
error in the principle of the simple pendulum moving in a circle. The period 
is almost but not quite independent of the excursion. This error does not 
exist if the suspension (Fig. 181) is modified so that the weight describes an 




FIG. 185. -THE VARIATION OF g WITH LATITUDE 
| = sin QOB = sin (90 - L) = cos L 

So if r radius of Latitude L (QB) and R the radius of the terrestrial sphere (OB), 
r = R cos L == cos L X 4,000 miles, 

e.g. at Lat. 45, since cos 45 = 0-707, r = 2,828 miles. The central acceleration at 
Lat. 45, when the units of distance and time are the foot and second, is 
(277) 2 r 40 X 2,828 X 5,280 A _ Q _ 
~TP~ ^ 24* x 60* x 602 =0-08 ft. per sec. 2 (approx) 

So (277 4- T) 2 r cos L is 0-08 x 0-707 = 0-06. Thus, if the value of g at the North 
Pole is 32-26, it is 32-20 (feet per sec. 2 ) at Lat. 45. 

arc of the curve called a cycloid. The cycloidal pendulum is isochronous. The 
quest for a perfect clock stimulated mathematical researches into the charac- 
teristics of new families of curves such as the cycloids and epicycloids. 



THE LAW OF THE SPRING 

The struggle for mercantile supremacy during the period which inter- 
vened between the two English revolutions of the seventeenth century was 

K 



290 Science for the Citizen 

accompanied by a lively interest in all scientific problems bearing on naviga- 
tion. Foremost among this English group of scientific men in the early 
days of the Royal Society and its parent body the Invisible College were 
Robert Hooke, Newton, Halley of comet fame, and Flamsteed, the first 
director of the newly founded observatory at Greenwich. Newton's studies 
on telescopy have been mentioned. Hooke's contributions to weather fore- 
casting will come later (pp. 557-9). Not the least important of his researches 
in the theory and practice of navigation were his various clock inventions. 
The principle of the stretched spring is still called Hooke's law. 

Hooke's position and personal characteristics are the portent of a new 
epoch in scientific discovery and the symbol of a past age. The socialization 
of scientific knowledge had taken a decisive step forward at the foundation 
of the new academies for the free distribution of new discoveries by regular 
publication in journals printed in the vernacular. Prominent among these 
was the Philosophical Transactions of the Royal Society. The need for new 
organs of education to replace or supplement the teaching of the medieval 
universities was already felt, though little was done to establish colleges and 
technical institutions free from ecclesiastical control until after the industrial 
revolution. In Britain, London, the hub of British mercantilism, was excep- 
tional. At Gresham College, the earliest forerunner of the London University, 
Hooke professed mathematics and mechanics. In the older universities 
astronomy had retained its traditional prestige, and medicine had gained a 
foothold (see pp. 557-9). There were now expanding opportunities of regular 
employment, and men of scientific capabilities were no longer wholly depen- 
dent on wealthy patrons if their social antecedents vouchsafed the necessary 
training. Regular employment for scientific pursuits and regular publication 
of discoveries went hand in hand. A new sense of common endeavour 
usurped the atmosphere of secrecy sustained by the insecurity of patronage 
and the dread of the Inquisition. 

Hooke's practice combined the generosity of the new age and the parsi- 
mony of the past. He was active in promoting schemes for co-operative 
endeavour, like his weather records, and lavish with fruitful suggestions 
(see p. 554). No doubt Newton reaped the benefit of some of them. He 
was never profuse in acknowledgments to Flamsteed, who supplied so many 
of his astronomical data, or to Huyghens or to Leibnitz with whom he corre- 
sponded. In contradistinction to this liberality, Hooke had a forgiveable if 
childish anxiety to wear the laurels of priority, and was one of the last 
scientific authors to use the time-honoured device of securing it by pre- 
liminary announcement as an anagram. He gave the law of the spring as the 
anagram ceiiinosssttuv two years before he disclosed the solution ut tensio 
sic vis in a published description of the experimental evidence for the law. 

Translated into the vernacular it means that the pulling power of a stretched 
spring is proportional to the displacement, a rule equally true for small dis- 
placements whether it is applied to elongation or lateral bending. It is easy to 
remember the rule when you know what it means, because the uniform 
graduation of a spring balance scale is an everyday illustration of its truth. 
Thus a weight of 3 Ib. stretches a spring 1 J inches if a weight of 2 Ib. stretches 
it 1 inch from its "natural" length, when no weight is attached to it. The 



Wheel) Weight^ and Watchspring 291 

stretch is therefore 5 inches per pound weight hanging from the spring, and 
the pound marks on the scale would be 0-5 inches apart. 

At first sight we might therefore draw the conclusion that the mass bears 
a constant, ratio to the stretch. This is not necessarily true. The pulling power 
of the spring is what prevents a weight at rest from falling. The string is 
stretched till the upward pull of the spring exactly balances the downward 
pull of the weight. If its mass is w, this pull is mg, and we already know 
that g is not exactly the same in all parts of the world. Provided the 
spring is not stretched too far, the ratio (m ~ x) of mass (ni) to distance 




FIG. 185A. A SEVENTEENTH-CENTURY BALANCE SPRING (BY PERMISSION OF PROF. 

WOLF) 

stretched (x) is fixed at any one place. It is not fixed for all places. What we 
should expect to find, and what we do find, is that the ratio mg -r- x is 
fixed, i.e. 

mg = kx 

The practical importance of this may be illustrated by the behaviour of a 
spring balance with a scale recording 2 Ib. per inch, at a place where g = 32 
feet per second per second. The extension (x) is ^ foot for a mass of 2 Ib., i.e. 

2 x 32 = k -f- 12 
.*. k = 768 (poundds per foot) 

If we took the same spring and the same weights to the top of a high 
mountain where acceleration under gravity is somewhat less, let us say 
31 feet per second per second, the extension would be less for each 
weight and the scale readings would not agree with the previous estimate. 



292 



Science for the Citizen 



The spring tension sufficient to balance a 2 Ib. weight would then be 2 x 31 
= 62 poundals, and since rng = kx, 

x = 62 -f- 768 (feet) 
= 0-96875 inch 

When we weigh with ordinary scales., we compare a body of unknown mass 
with a standard at the same place under the same pull of gravity. This is not 




= upward pull of spring 
+ 8 +16 +24 +32 



+4O 




FIG. 186. TESTING HOOKE'S LAW FOR A PIECE OF ELASTIC 

Up to 60 per cent extension an ordinary piece of rubber tubing gives a good result. 
The graph extension (x) plotted against the tension (F) is a straight line of which the 
slope is k, in this case 40 ~ T \ or 96 poundals per foot. 

true of a spring balance. A 2-lb. weight which produced an extension of 
1 inch at the bottom of the mountain would produce an extension of 0-96875 
(= 31 32) inch at the bottom. If the scale were marked at the bottom, 
a standard 2-lb. weight would register IT! Ib. at the top. So a spring 
balance like a pendulum clock is only reliable at the place where it was 
made, and the two devices provide independent methods for finding how g 
varies in different places. 



Wheel) Weight^ and Watchspring 293 



In certain circumstances, as we all know, a spring balance can be made 
to behave like a pendulum, as when a sudden jerk makes the load bob up 
and down. The fact that it does this illustrates in a new way the fruitfuhiess 
of measuring force by the acceleration imparted to a mass. If a mass of m 
hangs at rest on a spring (Fig. 187) stretched downward through a distance x 
beyond its natural length, Hooke's law tells us that its downward pull (mg) 
is balanced by the upward tension of the spring kx. If it is pulled through a 
further distance y and held, so that the total extension is now (x + y)> the 
total force exerted by hand and weight must be balanced by an upward 





< Periodic Timc(T)~ -> 

- Periodic Time(T) 



< 



FIG. 187. PERIODIC MOTION OF A WEIGHT AT THE END OF A SPRING 

The time graph is shown as it would be given by snaps at successive intervals of 
equivalent length. The equation can be built up from the mechanical model of Figs. 
] 82 and 195, which show that it has the general form (in circular measure): 

yt = r sin (2^1 -f- T) 

The constant r is the "amplitude" of the excursion, i.e. y, and the periodic time T 
is given in the text as 2-rrVx ~- g. Hence the equation of the time trace is 



The initial extension y does not affect the period. It only affects the excursion. 

tension of k(x + y)> When the hand is released, this upward pull is only 
offset by its own weight (mg kx) pulling it downwards. The latter is not 
sufficient to neutralize it. The difference between them is k(x + y) kx 
= ky. Hence there is still an upward tension numerically equal to ky, and the 
load begins to move upwards with an acceleration a. The force required to 
impart this acceleration to a mass m is ma, since the direction of motion 
(upward) is opposite to the direction (downward) along which y is measured. 
Hence 

ma == ky 

k 

/. a = --- v 
m J 



294 Science for the Citizen 

The acceleration is therefore in fixed proportion to the displacement y from 
the resting position of the weight, and of opposite sign to this displacement. 
In other words, the acceleration is always towards the resting position. 
Hence (p. 282) the motion is to and fro, and its period is 



2-77 Vm ~ k 
Since k = mg -~ x, this is the same as 



So, if as in the previous example a 2-lb. weight extended a spring through 
1 inch (jV foot) at a place where g 32, it would oscillate when jerked 



out of its resting position with a period 2 TT V> -f- 32 = 32 second 
(approximately) . 

A spring escapement of which the teeth engage a hanging weight would 
naturally have the same disadvantages as a pendulum, because the period of 
the osculation depends on g. The advantage of the spring is that it can be 
used to produce periodic motion without a weight hanging vertically. For 
instance, the end of a spirally wound spring like Hooke's hairspring of 
watches can be fixed to a flywheel, the inertia of which does not depend 
on gravity. The law of Hooke applies with remarkable accuracy to small 
lateral as well as to lengthwise strains. So this type of escapement provides 
an escape from the limitations of the pendulum clock. Hooke, like 
Huyghens, hoped to make a perfect marine clock. He did not succeed 
because the expansion of metals (p. 577) by changes in temperature was 
not sufficiently understood at the time. Navigation waited three-quarters 
of a century for what Newton called "a watch to keep time exactly." The 
social importance of the issue is shown by the Act of 1714, when the 
British Government offered a reward of 20,000 for any method to enable 
a ship to get its longitude with an error not exceeding 30 miles at the end of 
a voyage to the West Indies. Twenty years after the Board of Longitude was 
appointed to act as umpire, Newton's words were still true : "by reason of 
the Motion of a Ship, the Variation of Heat and Cold, Wet and Dry, and 
the Difference of Gravity in different Latitudes, such a Watch hath not yet 
been made." 

In 1736 a Yorkshire carpenter made a clock with a grid of brass and steel 
bars (see Chapter XI, p. 576) to compensate for variations in the tension of 
the balance springs at different temperatures. Harrison's first clock was 
tested in May of that year on a six weeks* voyage to Lisbon and back, making 
the outward journey on the Centurion, later Anson's flagship. The official 
certificate reproduced by Commander Gould in a centenary article discloses 
that Harrison located the Lizard correctly when the official navigator, Roger 
Willes, believed that the ship had reached Start Point "one degree and 
twenty-six miles" east of it. The invention was not offered for the prize. 
With small subsidies from the Board of Longitude Harrison persevered for 
more than twenty years. His fourth model in a test voyage to Jamaica in 
1761 led to an error of only one mile. The error was under ten miles in a 
second voyage to Barbadoes three years later. He was paid half the reward in 



Wheel, Weight^ and Watchspring 295 

1765 and only received the other half after a long legal quibble settled by 
a private Act in 1773. An exact duplicate of the fourth was made for the 
Admiralty at a cost of 450, and used by Captain Cook. A few years later 
Earnshaw, the inventor of the modern type, produced chronometers at less 
than a tenth of this price. 

UNIVERSAL GRAVITATION 

Artillery warfare first began to present new problems for research in the 
age of the great navigations, when astronomy was still the queen of the 
sciences. In one way or another inertia, the fundamental principle of range- 
finding, impressed itself on speculations about planetary motion before 
Galileo actually applied it to physical measurements. If a pull is necessary 
to keep a stone revolving at the end of a cord in its circular path, and if a 
bullet keeps its own motion in a straight line when the marksman is riding, 
how does it happen that the planets move in closed orbits round the sun? 
In an earlier age the question would have been meaningless. In the generation 
of Galileo there was a special reason for asking it. The mariner's compass 
was one of the new wonders of the age of the great navigations. Here for the 
first time was something which pulled without cords and pulleys. Here for 
the first time was "action at a distance." At the Court of Elizabeth the theory 
of the mariner's compass had as much news value as the Peking Skull in 
post-war England. The queen herself and her naval commanders gathered 
to watch experiments by Gilbert, the queen's physician, who had likened 
the influence of the sun on the planets to the earth's magnetism. 

Kepler had endorsed the analogy before Galileo showed how to measure 
pulling power by the motion it produces. The law of circular motion, which 
seems to have been discovered independently by Hooke and Newton as well 
as by Huyghens who first published it, made it possible to bring these 
speculations to earth. If a body moves with constant speed in a circle its 
acceleration along the radius is also constant and is equal to (27r) 2 r -=- T 2 . 
Kepler had shown that the ratio of the square of the time of revolution 
of a planet (or satellite) to the cube of the radius of its approximately circular 
path is the same for all planets (or for all satellites of the same planet), i.e. 

or ~=* 
If we combine both rules, 

(27r) 2 K 



That is to say, the acceleration along the radius of a planet's orbit is inversely 
proportional to the square of its distance from the sun round which it moves. 
Although this conclusion occurred to several people at the same time, 
Newton was apparently the first to test it. He argued that if the same pulling 
power draws a stone to the centre of the earth and keeps continually deflect- 
ing the moon from a straight path in its frictionless motion through empty 
space, the acceleration of the stone and the acceleration of the moon along 



296 Science for the Citizen 

the line joining it to earth must be in an inverse ratio to the square of their 
distances from the earth's centre. So he calculated the vertical acceleration 
of a body at the earth's surface from the moon's motion on the assumption 
that the same pull acts on both. The moon's distance r from the earth is 60 
times the earth's radius. So the acceleration of a body towards the earth's 

centre, if moving round the earth in the moon's course, should be ^ times 

\y\) 

the vertical acceleration of a body on the earth's surface. Since the latter is 
32 feet per second per second, it should require an acceleration of 32 -r 60 2 
feet per second per second to keep the moon bent in a circular course. A 
simple calculation is sufficient to show that this is correct. The mean distance 
(r) of the moon from the earth is approximately 240,000 miles or (240,000 x 
5,280) feet. The time (T) of a complete revolution is approximately 27 J days 
or (27 x 24 x 60 x 60) seconds. According to the principle of inertia, the 
acceleration required to keep the moon in its course is 

47T 2 47T 2 X 240,000 X 5,280 ft 

T^ ' r ^ 27? X 24* x 60* X 60* = 32 + 6 ( a PP ro *0 

The result obtained agrees with the assumptions that the rules which 
describe the regular motion of the clock wheels can also be used to calculate 
the motion of the heavenly bodies. When Newton first thought of making 
this calculation at the age of 23, the estimated distance of the moon was 
not very accurate, and the value he obtained was only seven-eighths of what 
he expected. Recognizing that reasoning by impeccable logic from self- 
evident principles is no substitute for solid fact in science, he locked away 
his calculations for over ten years. Meanwhile much discussion about the 
meaning of Kepler's laws had taken place among men of science. Hooke 
had arrived at the same conclusion as Newton, but was unable to solve 
the mathematical difficulties which arise when the orbit is elliptical, and 
the acceleration directed to the focus. A prize was offered for the solution of 
the problem by Sir Christopher Wren. Under pressure of his friends, Newton 
repeated his calculations with a new and more accurate estimate of the 
moon's distance based on the Cayenne expedition. The result was now 
satisfactory. So he was able to show that if the principle of inertia is true, 
a body can only move in an ellipse if the acceleration directed to its focus 
is inversely proportional to the square of the distance. He also showed that 
a body which moves in an ellipse with an acceleration directed to the focus 
must describe equal areas in equal times. 

Thus Kepler's three laws, which contradict our first impressions of the 
way in which the celestial objects appear to move, were brought into harmony 
with the experience of motion in everyday life. As with Hipparchus, so with 
Newton, experience of nature demanded new rules of reasoning. To solve 
all the problems which arose, Newton was compelled to improvise a new 
mathematical technique, the differential calculus. Contemporary idealist 
philosophers, notably Berkeley, poured the utmost contempt upon the new 
logical instrument. Today stubborn fact has triumphed over Berkeley's logic. 
The hostility of the metaphysicians has been long forgotten by a world 



Wheel, Weight, and Watchspring 297 

which prefers the comforts of science to the consolations of philosophy; 
and the belief that the rules of theory are exempt from the test of practice 
has been transferred from real science to political economy. 

Up to this point, Newton's contribution did little more than bring together 
the conclusions of his own contemporaries with more mathematical in- 
genuity than they had shown, and it is a falsification of history to look on 
Newton's theory as the production of isolated genius. His genius worked on 
problems which were set by the social circumstances of his time. In Galileo's 
treatment of the path of the cannon ball and in Huyghens' treatise on the 
clock the problem of motion in curved tracks had emerged from imperative 
social needs of the time. Again and again Newton's speculations turn to one 
or the other. In the elementary treatment of the mail bag given on p. 265 
and the analogous problem of a cannon ball (Figs. 171 and 172) fired in a 
horizontal direction, we assume that the height and range of projection are 
relatively small. We take the earth as approximately flat for our purpose 
and the acceleration of gravity as approximately constant, and directed 
towards the earth's surface. In an age of progress in artillery warfare it was 
natural to speculate further about what would happen if a projectile were 
fired at a great height with an enormous velocity. We then have to reckon 
with a variable acceleration directed to the earth's centre instead of an 
approximately constant g directed to an approximately flat surface. In the 
following passage Newton pictures the planets shot off from the sun at some 
time remotely past: 

That by means of centripetal forces the planets may be retained in certain 
orbits we may easily understand, if we consider the motions of projectiles. . . . 
The greater the velocity by which it is projected the farther it goes before it 
falls to the earth. We may thus suppose the velocity to be so increased that it 
would describe an arc of 1, 2, 5, 10, 100, 1,000 miles before it arrived at the 
earth, till at last exceeding the limits of the earth, it should pass quite by 
without touching it. ... If we now imagine bodies to be projected in the 
directions of lines parallel to the horizon at greater heights . . . those bodies, 
according to their different velocity and the different force of gravity in different 
heights, will describe arcs either concentric with the earth or variously eccentric 
and go on revolving through the heavens in those trajectories; just as the 
planets do in their orbs (see Fig. 188). 

Newton's special contribution was the next step. He had satisfied himself 
that the earth's attraction on bodies at its surface extends as far away as the 
moon. What was the nature of this pull which Kepler had likened to the 
action of a magnet? The sun pulled on the planets which are smaller than it 
is; the earth pulled the moon which is smaller than itself; and the earth's 
pull or terrestrial gravity is proportional to the mass on which it acts. So 
the pulling power of the sun on the planets might be connected with the 
fact that their masses are different. If so, it seems that every piece of matter 
exerts on every other piece of matter a pull directed to its centre and pro- 
portional to its mass. The mutual attraction between any two pieces of matter 
of mass m and M, separated by a distance r would therefore be proportional to 

m X M-f- r 2 
K* 



298 Science for the Citizen 

Putting this in the form of an equation, in which G is a universal constant 

F = G.^ 

If this is true, it is possible to calculate both the absolute mass of the earth 
itself, and the relative masses of the earth and other heavenly bodies. The 
first can be done in a variety of ways, one of which is shown diagrammatically 
in Fig. 189. How to find the ratio of the mass of the earth to that of another 
iheavenly body may be illustrated by comparing its mass with that of the sun. 

In Newton's calculation of the falling stone and the moon,, E is the earth's 
mass and the force on unit mass of either is GE ~- r 2 



Aioimfcain, 




FIG. 188. NEWTON'S PARABLE OF PROJECTILES FIRED WITH INCREASINGLY HIGH 

VELOCITY FROM THE TOP OF A VERY HIGH PEAK 

Note. The higher the initial velocity (see Figs. 171 and 172) the smaller the deflection 
towards the earth's centre in a given distance traversed. By increasing the velocity 
of projection the motion therefore approaches a closed elliptical orbit, such as a rocket 
projected beyond the stratosphere would pursue in empty space. Air resistance is 
neglected. 

Since F = ma, the acceleration g e of the stone at the earth's surface, and 
the earthward acceleration, g m , of the moon, are in the ratio r^ : r' 2 e , and the 
earth's mass cancels out. The problem is different when we compare the size 
of the earth's orbit about the sun with the size of the moon's orbit about 
the earth. We then have two different masses: S, the sun's pulling on the 
earth at a distance R; and E, the earth's pulling on the moon's M at a 
distance r. The attraction of the earth to the sun is 

G . SE -T- R 2 

The attraction of the moon to the earth is 

G . EM ~ r 2 



Wheel, Weight, and Watchspring 299 



So the ratio of the two attractions (earth to sun) : (moon to earth) is 

Sr 2 -f- MR 2 
If A is the acceleration of the earth sunward along the radius of its orbit. 



3 feat to centre 








" 'attractive 
force, of 
lead hall 



FIG. 189. WEIGHING THE EARTH 

The principle of weighing the earth by the deflection of the plumb line towards the 
mountain is the same as that used in the direct determination of the mass of 
the earth by Cavendish and later workers who have used a sort of microscope 
to measure the deflection of a suspended pellet Q towards a large mass. 
Gravity acts towards the centre of a spherical mass. If a pellet is deflected through 
a from the vertical when placed near a large mass, like a ton weight of lead, the ratio 
of the attractive forces of the lead and the earth itself is tan a (see legend to Fig. 167). 
According to Newton's hypothesis of universal gravitation, the attractive power of a 
body is directly proportional to its mass and inversely proportional to the square of 
the distance between its centre of mass and the body attracted by it. Hence if m is 
the mass of a body whose centre is r feet from the pellet, and M is the mass of the 
earth whose radius is R, 

mR 2 
tana = 



Suppose in such an experiment (crudely diagrammatized in this figure) 
a 0-00000045 (i.e. tan a = 8 x 10~ Q ) when m = 1 ton and r = 1 yard or TT V 
miles. 



x 10-9 



M 



4,000 2 x 1,760* 
M 



1 x (4,000) 2 
M X (1 ~ 1,760)* 
16 X 176 2 X 10 8 

8 X 10- 
6* 5 x 10 21 tons approximately. 



the force pulling the earth to the sun is EA. And if Y is the time of a complete 
revolution of the earth round the sun, 

47T 2 

EA = E . -~ . R 



3OO Science for the Citizen 

GSE 



R* 

Similarly if L is the time of a complete revolution of the moon round the 
earth 



Hence the ratio of the sun's mass to the earth's mass or 
S R 8 L 2 (93,000,000) 3 x ^.7^2 



E r 3 Y 2 (240,000) 8 x (365) 2 



= 340,000 (approximately). 



The best determinations give 333,434. The large mass of the sun, 
compared with that of any of the planets, is the reason for the fact 
that they all appear to describe orbits around the sun as focus. How- 
ever, their orbits do not absolutely correspond to Kepler's laws, because 
they exert attractions on one another. The great triumph of Newton's theory 
of universal gravitation was the fact that such irregularities can be calculated, 
when the masses have been estimated in some such way as the example given 
illustrates. Just as the planets exert minor effects on the motion of their 
neighbours, the sun, which is too far from the moon to affect the more 
obvious characteristics of its motion, produces various irregularities which 
had been quite unintelligible. From the ratio of the mass of the moon and 
the sun, and their distances from the earth, Newton was able to account 
for the variations of the tides with the phases of the moon (Fig. 190), for 
the inclination of the moon's orbit and the regression of its nodes. 

It is natural and proper for you to ask at this point whether Newton's 
theory provides any guidance for social practice. One, directly related to the 
focal problem of technology in the social context of Newton's time, is the 
increase in the period of a pendulum or the apparent decrease of a mass 
weighed in a spring balance when we ascend a mountain. If the law of inverse 
squares is right the value of the earth's pulling power on unit mass (i.e. g) 
must increase as we get nearer the earth's centre and decrease as we get 
farther away from it by a calculable amount. Taking the earth's radius as 
4,000 miles at the surface, the ratio of g at the surface to g one mile above it 
at the top of the mountain should be (4001) 2 : (4000) 2 , and if g were 32 at 
the bottom it would be 31 984 at the top. Hence at the bottom the period of 
a pendulum would be 2?rVL -f- 32 and at the top 277 VL -f- 31-984, and 
the periods at the bottom and top would be in the ratio V31-984 ~ V32. 
The clock would lose time in this ratio per second.* 

* Huyghens rightly assumed that differences of sea-level measurements of in 
different latitudes is not due to irregularities of this kind. The perfect sphericity of 
the earth was an accepted dogma, sufficiently justified by later measurements in low 
latitudes 



Wheel, Weight, and Watchspring 301 

The application of Newton's theory was first made by Bouguer in one of 
the great scientific explorations of the eighteenth century. Bouguer com- 
pared the lengths of die seconds pendulum at sea-level and at the top of 
Pinchincha, a high mountain above Quito in Ecuador. The necessary 
reduction for the ascent calculated by the inverse square law was ri A r> 




VEpm 

(low 

water) 




(high water) 

TEBT ldbON(Ugh tide at noon) DAY AFTER XEft 7 MOON" 
tuiz 15 a ZittZe ktcr each d^r 



W^W, 





Wew 



Moon 

. occur at New or Full Moon, 
puZL of 5toL & Tvfcorz, ax ttigeilier 
----- L era^lopa. 



-->(Eartfi 



FtiZi 

-o 

Moon 



iid^s o<xiir a tlie first 6 
Quarters, when th& pull of the Sun 
partly nzudralises that of the Moon 




FIG. 190. THE TIDES 



this proved to be larger than the observed value j VT. What seemed to be a 
disappointing result led to a new vindication of the Newtonian theory. 
Bouguer argued that a large mass such as a mountain must exert an appre- 
ciable sideways attraction. This was triumphantly confirmed by a simple 
device. He selected two stations on the same parallel of latitude, one close to 



302 Science for the Citizen 

Chimborazo, a mountain 20,000 feet high, and one some distance due west- 
ward, away from the influence. The inclination of a telescope to the plumbline, 
i.e. the zenith distance, should have been the same at meridian transit since the 
latitudes were identical. Owing to the attraction exerted by the mountain on 
the weight of the plumbline the zenith distance of the star at the near station 
was greater than it should have been. From this observation Bouguer calculated 
the density of the earth, and figures obtained in this way agree fairly well with 
results obtained by the direct method of Fig. 189. 

In the next chapter we shall see how the invention of an instrument 
which could measure time in very short intervals had more far-reaching 
results than any which we have discussed so far. One result of this was 
unexpected confirmation for the belief in the earth's orbital motion about 
the sun. Before leaving the story of clock-making we shall now summarize 
some terrestrial experiences which lead us to believe that the apparent daily 
motion of the celestial sphere results from the earth's axial rotation. Two 
have been mentioned, namely: (a) that pendulum clocks lose time as we 
travel from the poles to the equator, while spring clocks do not do so; (V) that 
the earth and other heavenly bodies are flattened slightly at the poles like a 
ball of soft clay on the potter's wheel. Three others will now be mentioned. 

EARTH'S DIURNAL ROTATION 

If a projectile is shot in any direction from the North Pole, it continues 
with uniform horizontal velocity while the object at which it was aimed is 
being carried round leftwards, owing to the counter-clockwise motion of the 
earth as seen from the standpoint of a polar observer. Hence the bullet will 
fall right of its mark. Conversely a projectile fired from the South Pole 
would fall left of its mark. If the projectile were fired at an object south of 
the equator from a situation at the same distance from the equator north of 
it, there would be no such displacement. In any other situation a bullet fired 
north or south would deviate slightly from its mark because the east-west 
velocity of the object and projectile would be different. The effect would be 
negligible if the distance between them were small. With long-range modern 
projectiles the distance is sufficiently large to make appreciable errors in 
marksmanship if it is neglected. In actual practice, the error is avoided by 
making an easily calculable allowance for the earth's axial motion. If rocket 
transmission, which is already used for postal services in a few remote places, 
became general, the reality of the earth's axial motion would become an 
ever-present feature of social communications. In his fascinating book, 
Rockets Through Space., Cleator tells us : 

After conducting a series of preliminary experiments, Schmiedl succeeded 
in establishing, in 1931, an officially recognized rocket postal service. He 
operated his service between the small towns of Schockel and Radegund, near 
Graz, Austria. Although the distance covered was only about two miles, the 
mountainous nature of the district enabled him to transmit letters from one 
town to the other in as many minutes as the ordinary postman required hours. 
The success of the Schmiedl service inspired Gerhard Zucker to perform a 
like experiment in Germany. And in 1933 he successfully transported letters 



Wheel) Weight^ and Watchspring 303 



by rocket over the Harz mountains. A year later, as is well known, he brought 
one of his rockets to England where his experiments were a fiasco. . . . 
Experimenters visualize the time when mail rockets will carry their cargo 
between London and New York, a distance which even the liquid fuels of 
today will enable the vessel to cover in less than an hour. 

Another consideration is not so obvious. If a pendulum bob is suspended 
from a very long cord (e.g. 100 feet), free to rotate in any direction and set in 
motion, it should continue to swing in the same direction. If the direction of 



After 470 days / 
on 'Vetw.s / 



146 ioy 




Earth, , 

of trip 



to Earth, after 
a "half jourruzy round 
Swi , J762 days after 
starting^ 



FIG. 191. WILL THEORIES OF GRAVITATION BECOME NECESSARY FOR TRAVEL? 

The figure shows Dr. Walter Hohmann's project for a space trip by rocket transmission 
out of the first 50 miles of the earth's atmosphere. Beyond this distance air resistance 
becomes negligible and the space ship behaves like a planet moving in a curved path. 
The trip would take 762 days, including a stay on Venus of 470 days, or more than 
two Venus (225-day) years. The present cost would be about 20,000,000. Considering 
that we have not yet liquidated the social inconveniences resulting from the African 
slave trade which followed the discovery of America, this is not necessarily a dis- 
couraging fact. 

the swing is marked by a line on the ground, the pendulum swing will appear 
to twist slowly around it in a clockwise direction (in the northern hemisphere). 
No force has been applied to the pendulum to change its direction., and we 
ought therefore to say that the line has twisted in an anti-clockwise direction 
beneath the pendulum. You will have no difficulty in seeing that this is a 
necessary consequence of the earth's axial rotation if you suppose the pen- 
dulum to be suspended at the North Pole, in which case the line will perform 
# complete anti-clockwise rotation in 24 hours. At the equator the Hpe >vilj 



304 



Science j or the Citizen 

c 




FIG. 192. WHY THE PLANE OF THE PENDULUM SWING ROTATES 

Suppose that the pendulum is started swinging in the meridian at a, i.e. along the tangent 
ac, which meets the axis OP produced at c. A short time later the earth has rotated 
through a small angle aob, carrying the pendulum to the point b on the same parallel 
of latitude. The tangent be meets OP in the same point c. The pendulum is, however, 
still swinging in the direction ac, i.e. along bd } so that relative to the meridian it will 
apparently have changed its direction clockwise through the angle cbd = bca. Mean- 
while the earth has rotated through the angle boa. The rates of rotation are proportional 
to the angles turned through in the same time. Hence 

rate of rotation of pendulum /_bca arc ab . arc ab 



rate of rotation of earth 



/Lboa 



. 

_ sin aco sm 

ac 



sin Lat. 



So the time of rotation of pendulum = ~ days. 
* sin Lat. J 

Thus in Foucault's experiment the pendulum swing rotated through 360 in 32 hours 

or 1J days. Since the latitude of Paris is 48 50 ' 3 -. - = = H. 

J sin Lat. 0-75 (approx.) 

not rotate. At the South Pole the line will rotate completely in a clockwise 
direction in 24 hours. At intermediate latitudes, the time of rotation of the line 

is ~ - days, as shown in Fig. 192. A large pendulum suspended from a 
sin Lat. 

high roof above a dial was exhibited at the Paris Exhibition by Foucault to 
show this phenomenon to the public. Foucault's pendulum is now shown 
in one of the great museums of the U.S.S.R. 



Wheel., Weight, and Watchspring 305 




FIG. 193. THE GYROCOMPASS 

The framework of a gyrostat turns freely at A, and round the axes BC and DE, so that 
the axle FG can point in any direction in space. If no force is applied to the frame- 
work, the axle FG will continue to point in the direction in which it was set when the 
flywheel was started If, however, a turning force is applied to the framework, the axle 
turns, not in the plane of the force, but in a plane at right angles to it and continues to do 
so, until the plane of rotation coincides with the plane of the turning force. This is why 
the gyrocompass turns to the north. If, for example, the axle FG is at first pointing 
eastwards, it will tip upwards as the earth rotates. If the framework is provided with 
some kind of ballast (like the mercury tube in the figure) this will now exert a turning 
force on the framework. The axle will therefore turn at right angles to the plane of the 
force, until it points to the north. The axle will then return to the horizontal, and 
the turning force will cease to act. 



3 o6 



Science for the Citizen 





IcpnafftnaL 
. 'Magnetism. 

Gilbert- (1600) 



PRINTING 

Block books, 14* cent. 
Tvfovabieiype, 3453... 



SPECTACLES (I4*cen$ | 



\rELESCOPE(l6U) 



Wheel, Weight, and Watchspring 307 



WE5TERLY 

uo tiu Atlantic 
LX3N&ITUDE at 




Keplzr(l609-19)- 



Copernicus, Ijg 
" 



IZNZWEBSai GRAVITATION. 

Newtons Prindpia.(16&7) 

A 



CLOCK? 

Observatories 
(1470 onwards) 



PLANETARY SATELLITES 

CassifiL ^ Primer (1650-'/0) 




Pendulum 

(about 1600) 




SPEED Of 

LIG-HT 
Earner (16*76) 



t 



SPEED OFSOZZND; 

iYlr32TUl&, t w<35S27iClL 

Caiout ^640) 



\ 



STRIKED INSTRUMENTS 
(with keyboard,) 



TIME CHART II 



3O& Science for the Citizen 

Perhaps the most striking phenomenon which illustrates the earth's axial 
motion is the modern gyrocompass now used as a substitute for the magnetic 
compass, which is no longer suitable because modern ships are made of iron 
and carry electrical machinery. It is essentially like the familiar gyrostatic 
top which consists of a heavy metal wheel mounted on two concentric rings 
with their axes at right angles (Fig. 193), so that it can rotate in any plane. 
Anyone who has played with a gyroscopic top will have found that it resists 
attempts to change the direction of the axis of rotation, and turns instead 
in a plane at right angles to the plane of the applied turning force. If therefore 
the axis of rotation of a simple gyrostat is set towards any star it continues to 
stay so, while its orientation to the earth's surface changes in accordance with 
the earth's position relative to the star, i.e. as if the stars were fixed relative 
to the earth's axial motion. The gyrocompass (Fig. 193) is an electrically 
driven gyrostat, provided with some form of ballast, so that a turning force 
comes into play when the axle tips owing to the earth's rotation. The axle 
therefore turns until it points in a direction at right angles to the plane of 
the turning force provided by the ballast, i.e. horizontal and to the true north. 
The gyrocompass has thus the double advantage that it points to the true 
north in contradistinction to the variable magnetic pole, and is not affected 
by the magnetism of the steel plating used in the construction of a modern 
ship. 

THINGS TO MEMORIZE 

1 . The Lever. Where D and d are distances from the fulcrum, and H and h 
are heights through which weights W and w are moved, WD = wd and 
WH - wh. 

o T i- , , Weight hanging vertically . . . . 

2. Inclined plane. ~ - ?~~ - = sin (angle of slope). 

Balanced weight on slope 

3. Motion with constant acceleration. If the initial velocity is w 3 d = ut -f- %at 2 
or, where motion is from rest, d = \a&. If the body rolls down a slope, 
a = g sin by where b is angle of slope. If it falls vertically, sin b == 1 5 so 
a = g = 32 ft. per sec. 2 , and d = 16r 2 ft. See also the four rules on p. 258. 

4. Force = mass x acceleration. 

1 poundal = force required to impart accel. of 1 ft. per sec. 2 to mass of 1 Ib . 
1 dyne = force required to impart accel. of 1 cm. per sec. 2 to mass of 1 gm. 

5. Motion in a circle. Accel, towards centre (27rw) 2 r. 

6. Motion is periodic when: Accel = kx, where x is displacement along 
a fixed line from a middle position. Both measured to the right. 

The period, T = -^L 

Vk __^ 

7. Simple pendulum. T = 2<7r\/ 

v g 

8. Kepler's Laws. I. Every planet moves in an ellipse, with the sun in one 
of the foci. 

II. The straight line drawn from the centre of the sun to the centre of the 
planet sweeps out equal areas in equal times. 



Wheel) Weight, and Watchspring 309 

III. The squares of the periodic times of the several planets are proportional 
to the cubes of their mean distances from the sun, i.e. r 8 = KT 8 . Hence accel. 

along the radius of the orbit = a . 

9. Law of gravitation. F = G , where G is a universal constant. 



EXAMPLES ON CHAPTER V 

Take g as 32 ft. per sec. 2 , unless another value is given 

1. Find the velocity at which a man must swim across a river 140 yards 
wide, if the current velocity is 2 miles an hour, and if he does not wish to 
be carried farther down the river than 40 yards. 

2. A marksman in an express train moving at 60 miles an hour sees a 
stationary object 100 yards away on a line at right angles to his line of motion. 
How far to one side of the object must he aim if he wants to hit it, the bullet 
moving with a velocity of 440 ft. per sec.? 

3. What is the resultant velocity of a ship heading in a northerly direction 
at 5 miles per hour, but drifting westward with the tide at 3 miles per hour? 

4. A trawler is steaming due north with a velocity of 15 miles an hour 
and a north-east wind is blowing with a velocity of 10 miles an hour. Show 
by a diagram the direction of the smoke track. 

5. At a certain time one ship is 10 miles west of another. If the first sails 
north-east at 20 miles an hour, and the second north-west at 16 miles an hour, 
how near do they approach? 

6. A stick 100 cm. long weighs 92 gm., and balances on a knife-edge at 
50-2 cm. along it. Where will the stick balance if a 50 gm. weight is slung on 
it by a piece of thread at the 10 cm. mark? 

7. At what distances from the fulcrum can weights of 20, 40, 80, and 150 Ib. 
balance a 60-lb. weight attached 1 foot 8 inches from it on the opposite side? 

8. What is the force exerted by a spring balance attached by a piece of 
thread to a metre stick, weighing 92 gm., at 90 cm., when the stick is resting on 
a wedge at 10 cm. and carries a 50 gm. weight at 60 cm.? 

9. Making no allowance for the weight of a 4-ft. crowbar, what force 
(poundals) must the arm apply to one end of it to lift a weight of 1 cwt. (1 12 Ib.) 
when a stone is placed under the crowbar 6 inches away from the end on which 
the load rests? If the load is 100 kilos give the force in dynes. 

10. A pole of 5 yards and of negligible weight has weights of 7, 5, 3 and 1 cwt., 
at 1, 2, 3 and 4 yards respectively from the end where it is hinged. What pulling 
power in hundredweights must be exerted upwards at the free end to sustain 
the weights? What is the upward pull exerted by the hinge? 

11. If a 4-ft. crowbar weighing 20 Ib. is suspended by a cord 2J feet from 
one end, find what weight placed 1 foot from one end will balance a 5-lb. 
weight at 9 inches from the other, taking into consideration the weight of the 
crowbar itself. 

12. A pole, whose weight can be neglected, rests on the shoulders of two 
men. If the maximum weight one man can carry is 120 Ib. and the maximum 
weight the other man can carry is 90 Ib., how heavy a man can they carry 
between them, and where must he balance on the pole from the stronger man? 

13. A weight of 50 Ib. rests on a smooth surface. What weight suspended 
vertically will balance it, if the surface is tilted through 10, 30, 45, 60, 
and 90? 



3io Science for the Citizen 

14. The ends of a cord are stretched over two pulleys with weights of 3 and 
41b. respectively. If a weight of 5 Ib. is attached to the cord between the 
pulleys what will be the angle at the point of suspension? 

15. Making no allowance for friction, tabulate the acceleration of a billiard 
ball down a smooth slope tilted successively from to 90 in intervals of 10. 

16. What are the distances traversed in 1 second, and the velocities acquired 
in that time, by smooth balls rolling down smooth slopes of inclinations 30, 
45, 60? 

17. Making no allowance for friction, find (with scale diagram or tables of 
sines) : 

(a) How long it will take a billiard ball to traverse 1 yard along a slippery 
slope at 60 to the ground. 

(b) How far the ball will move along a slope tilted at 15 in 5 seconds. 

(c) The speed of a ball which has traversed 2 feet on a slope tilted at 10 
upwards. 

(d) The speed of a ball after moving for 3 seconds on an inclined surface 
at 5 to the ground. 

In each case the ball starts from rest. 

18. A billiard ball starting from rest on an inclined glass surface describes 
40 feet in the third second. What is the inclination of the plane? 

19. If g is 32 ft. per sec. per sec. at a certain place, what is the value 
of g in cm. per sec. per sec.? If the metrical unit of force called the dyne is 
exerted on 1 gram to produce an acceleration of 1 cm. per sec. per sec., how 
many dynes correspond to (a) one poundal, (b) one pound weight suspended 
vertically? (1 kilogram ==2-2 Ib.) 

20. What is the mass in pounds of a body, if an 8-lb. weight, hanging ver- 
tically from a pulley, acts on it for 3 seconds, raising it with a final velocity of 
6 ft. per sec.? 

21. What force in tons weight dragging horizontally from a train of 200 tons 
mass going at 30 miles per hour will stop it (a) in one minute, (6) at a distance 
of 400 yards? 

22. What is the stretching force in the cable if a lift weighing one ton 
descends with an acceleration of 16 ft. per sec. 2 ? With what force does a mass 
of 8 Ib., lying on the bottom of the lift, press on the latter? 

23. Draw the speed time diagram of the motion of a train starting from 
rest, having uniform acceleration for the first 15 seconds of its run, running 
i minute at a constant speed, and then coming to rest in 7| seconds with 
uniform retardation. What is its maximum speed if the distance traversed is 
2,420 feet? 

24. If a train is brought to rest in If minutes, with uniform retardation, 
when it is moving at \ mile per minute, how far has it travelled in this time and 
what is the retardation in feet per sec. per sec.? 

25. If the initial velocity of a body moving in a straight line is v and its 
velocity after moving d feet in t seconds with a constant acceleration a is v t3 
show that the distance moved can be determined from the formulae : 

(i) d = v + % at2 
(ii) d = (02 - vg) ~ 2a 
(iii) and when the initial velocity is zero v t = V%ad 

26. How high will a ball rise, and how long will it take to reach the highest 
point, if it is thrown vertically upwards with a velocity of 2 3 760 yards per 
minute? 



Wheel, Weighty and Watchspring 311 

27. A bullet moving in a straight line has a velocity of 5 yards per second, 
and an acceleration of 10 feet per second per second. How long will it be before 
the velocity is tripled and how far will the bullet travel in this time? 

28. A ball is thrown vertically upwards and rises to a height of 10 yards, 
Find the velocity the ball must have been given and the time it takes to reach 
this height. 

29. How far must a ball fall from rest to gain a velocity of 20 feet per second? 
How much farther must it fall so that the velocity is increased to 30 feet per 
second? 

30. At what speed does a train moving with uniform acceleration run at the 
end of 4 minutes from starting up, if it traverses exactly one mile in the interval? 
Find also its acceleration in feet per sec. per sec. 

31. A cannon ball is fired horizontally from a 49-foot tower, with a muzzle 
velocity of 2,000 feet per second. Neglecting friction, calculate the distance 
from the tower at which the cannon ball will hit the ground. 

32. A bomb is dropped on a gasometer by an aeroplane flying horizontally 
1,000 feet above it. If the aeroplane is travelling at 60 miles per hour, what is 
the angle between the vertical and the line joining the bomb and gasometer 
when the bomb is released? 

33. If the muzzle velocity of a cannon ball is 420 feet per second and the 
range is 1,800 yards, find the angle at which the cannon is tilted. 

34. A cannon ball is projected at an angle of 30, with a velocity of 192 feet 
per second. When will it reach a height of 80 feet above the ground, and how 
far from the cannon will it be at that instant? 

35. (a) If a cyclist takes a corner at 10 miles per hour in a curve of 5 yards 
radius, what is his acceleration towards the centre of curvature? (fc) Find the 
central acceleration of a point on the equator in feet per sec. per sec. Take 
the earth's equatorial radius as 3,962 miles and the sidereal day as 23 hours 
56 minutes 4 seconds. 

36. A motor cyclist goes round a circular racecourse at 120 miles per hour. 
Find how far from the vertical he must lean inwards to keep his balance (a) if 
the track is 1 mile long, (b) if it is 880 yards long (see Fig. 180). 

37. A locomotive goes round a curve, whose radius of curvature is \ of a 
mile, at 30 miles per hour. If the rails are 4 feet apart horizontally, how much 
higher must the outer rail be, so that the engine wheels exert no pressure 
outwards on it? 

38. By counting the beats over two minutes, test the formula for the period 
(p. 287) of a pendulum by attaching a button to a piece of cotton 1 foot, 2 feet, 
3 feet, 4 feet long, suspended by a loop from a drawing pin. 

39. What are the lengths of simple pendulums which make complete beats 
(forwards and backwards) in (a) 14 seconds and (fc) in 24 seconds? (Take g as 
32 feet per sec. per sec.) 

40. What is the acceleration due to gravity at a place where a simple 
pendulum, of length 37-8 inches, makes 183 beats (half periods) in 
3 minutes? 

41. Show that a pendulum, 450 feet long, makes a complete beat in about 

23f seconds. 

42. What is the length of a simple pendulum, if on shortening it by 1 inch 
the period is diminished by ^> of its value? 

43. A seconds pendulum at a place where g is 981-4 cm. per sec. 2 is taken 
to a place where g is 981 -0 cm. per sec. 2 . Find how much it gains or loses in 
a day. 



312 Science for the Citizen 

44. A simple pendulum, which should beat seconds (i.e. one whose half 
period is one second) loses 20 minutes in a week. How much per cent must its 
length be shortened to make it keep time? 

45. A pound weight is attached to the end of a piece of string 1 yard long. 
What stretching force would be registered by a spring balance, when the 
weight is swung in a horizontal circle so that it takes | second to make one 
complete circle? 

46. A wire of 497 cm. is stretched 35 cm. by a weight of 6 kg. How much 
would it be stretched by i cwt. and by 1 Ib. if Hooke's law applies over the 
whole range? (1 kg. = 2-2 Ib.) 

47. If a weight of 1 Ib. oscillating at the end of a spring executes a complete 
movement up and down in half a second, find the extension per Ib. 

48. If a weight of 250 grams executes a periodic motion with a complete 
period of 1J seconds at the end of a metal spring, what would be the period of 
oscillation of a 160-gram weight attached to the same spring? 

49. If F^ is the sea-level centrifugal acceleration at latitude L due to the 
earth's motion and # 90 is the acceleration of a body under gravity at the pole, 
show that the corresponding acceleration under gravity at latitude L on a 
spherical earth is 

90 - F/, . cos L 

With a similar figure show that if R is the radius of the equator plane, the 
radius of the latitude circle L is R cos L. Hence, taking R = 4,000 miles, 
calculate the values of g at latitudes 50, 45, and 30 in ft. per sec. per sec. and 
cm. per sec. per sec. 

50. At one place A the value of g is 983-0 cm. per sec. per sec. At a second 
place B it is 981-0 cm. per sec. per sec. One spring balance is calibrated at A 
and another at B. What will be the weight of a standard 10 Ib. mass indicated 
on the scale of each balance at each place? (g at London = 981-17). 

51. If the force of terrestrial gravity varies inversely as the square of the 
distance from the earth's centre, what would be the difference between the 
value of g at sea-level and at the top of a peak 5 miles high? How would a spring 
clock and a pendulum clock which synchronize at sea level behave in the two 
situations? 

52. A pendulum clock with a seconds pendulum* is carried in a balloon, 
which is ascending with a constant acceleration, to a height of 900 feet in 
1 minute. Show that the clock gains at the rate of roughly 28 seconds in an 
hour. 

53. A seconds pendulum beats 60 J times in 1 minute in a train, which is 
moving uniformly round a curve at 60 miles per hour. Show that the radius of 
the circular curve must be roughly 1,317 feet. 

54. At the top of a mountain a seconds pendulum loses 10 seconds in a day. 
How high is the mountain, and how many seconds would the pendulum lose 
when only halfway up the mountain? 

55. How high is a mountain if a clock, which gains 10 seconds a day, is taken 
up it and is found to lose 10 seconds a day? What is the difference in the 
accelerations due to gravity at the top and bottom of the mountain? 

56. With the aid of Fig. 185, tabulate the radius of the latitude circle and 
the speed of rotation in miles per hour at Aberdeen 57 N., Edinburgh 56 N., 
London 51|N., Falmouth 50 N., Paris 49 N., Philadelphia 40 N., New 
Orleans 30 N and Cayenne 5 N taking the equator radius as 4,000 miles. 

* A seconds pendulum has a half period of one second 



Wheel, Weight, and Watchspring 313 

57. Taking the value of g as 32-26 (ft./sec. 2) at the North Pole, use these 
measurements to calculate for each place at sea-level. 

(a) the value of g m , 

(>) the length of the seconds pendulum in inches. 

58. Use the same data to calculate the daily retardation of a pendulum clock 

(a) set at Aberdeen and sent to Falmouth; 

(&) set at Paris and sent to Cayenne; 

(c) set at Philadelphia and sent to New Orleans; 

(d) set at Edinburgh and sent to London. 

59. Using the same data calculate how slow or fast a London pendulum 
clock would be if taken to the top of Mt. Everest (5 miles high, Lat. 28 N.). 

60. If a man standing at the North Pole fired a bullet with a range of 3 miles 
and a mean horizontal speed of 400 feet per second at an object 3 miles away, 
how far wide of the mark would it fall and on which side? 

61. If a cyclist is riding at 14 miles an hour with a wind blowing at right 
angles to his path with a velocity of 10 miles an hour, show by a diagram or 
otherwise the direction of the wind relative to the cyclist. 

62. A man is walking due west with a velocity of 5 miles an hour. If another 
man rides in a direction 30 west of north, what is his velocity if he always 
keeps due north of the first man? 

63. A north wind is blowing at 10 miles an hour. If to a cyclist it appears to 
be an east wind of 10 miles an hour, find the direction and the rate at which he 
is riding? 



CHAPTER VI 

THE SAILOR'S WORLD VIEW 

The Wave Metaphor of Modern Science 

WE can only find our way about in space when we have learned to find our 
way about in time., and the accuracy with which we can ascertain where we 
are depends on the accuracy with which we determine the time when we 
happen to be there. Our own generation has witnessed a revolution which may 
prove to have more far-reaching results than the invention of the clock. A 
ship's captain is no longer dependent on a chronometer set by Greenwich 
time at the beginning of a voyage. Standard time is transmitted from ocean 
to ocean by wireless signals, and the very word wave has come to have a new 
meaning in the everyday life of mankind. Today the complete narrative of 
man's conquest of time and space would have to tell how it has become 
possible to put any single being in instantaneous communication with anyone 
else on the planet which we now inhabit. At some future date it may even 
record how human beings learned to find their way across interplanetary 
space. We must leave the way in which man has established the means of 
world communication till we come to the story of man's conquest of power. 

Before we do so, we shall have to come to grips with one of the most difficult 
concepts of contemporary science. The invention of wireless transmission 
was made possible by the theory of wave motion, which developed as a 
by-product of Newtonian science. In man's earliest attempts to find his 
bearings in time and space the nature of light had already forced itself on 
his attention. The discovery of the telescope gave a new impetus to the 
study of optical phenomena. One result of this was Newton's discovery 
that white light is complex. Very soon after the invention of the telescope a 
Danish contemporary of Newton made the first telescopic maps of the moon's 
mountains. Romer also took up the study of Jupiter's satellites. Having 
determined their period of revolution when Jupiter and the earth are on the 
same side of the sun, he calculated their position when Jupiter is on the side 
opposite to the earth. An eclipse of one of Jupiter's moons occurred later than 
he expected. Being satisfied that this was not due to faulty instruments he 
concluded that light does not travel instantaneously. To put it less meta- 
phorically, we do not see an event when it happens. A measurable interval 
of time elapses between a flash of light and the instant when we see it. This 
interval depends, like sending a message, on the distance between the sender 
and the receiver. 

At first sight it seems a far cry from the moons of Jupiter to the everyday 
life of mankind in seventeenth-century Europe. In its own context Romer 's 
work was not so remote from practical application as it would have been 
if it had been undertaken a century later, when clock technology had reached 
a higher level. The determination of longitude still remained a thorny 
problem. Judging from the prize they offered (p. 294), the method of lunar 



The Sailor's World View 315 

distances which Newton advocated with appropriate corrections for parallax 
and atmospheric refraction required laborious calculations which did not 
commend themselves to the Admiralty. Others among Newton's con- 
temporaries proposed to discard astronomy and use a co-ordinate based on 
the dip of the magnetic needle (see Chapter XIII). Failing a reliable chrono- 
meter, which was not available till 1760, the astronomer could only retain 
his privileged position as doyen of nautical science by devising a method 
as simple as the observations of eclipses or occultations. The only serious 
objection to this method is that celestial signals known before Galileo's 
observations on the moons of Jupiter did not occur very often. The objection 
could be met if other more frequent signals could be added. Whitaker's 
Almanack shows that fifty-six appearances or disappearances of Jupiter's 
satellites could be seen at Greenwich after dark in 1936. There were only 
two visible eclipses and no planetary occultations in the same year. 

Galileo himself seems (see p. 134) to have realized the practical import- 
ance of the discovery which brought him into conflict with the Inquisition. 
At the new Paris observatory where Romer worked with Picard, Richer, 
and others among the leading astronomers of the day, the Italian Cassini 
undertook the preparation of tables for calculating longitude by observations 
on the satellites of Jupiter, and his tables were used in the French Navy 
during the first half of the eighteenth century. In his book Histoire de la 
Longitude de la Mer au XVIHe si&Ie en France,, Marguet tells us : 

After 1C 90 the Connaissance des Temps gave the time of eclipses of the first 
satellite, calculated according to the tables of Cassini, and forty years later, 
from 1730 onwards, there were added to the ephemerides three other small 
moons of Jupiter known at this period. 

The determination of longitude by eclipses of Jupiter's satellites merely 
depends on the known fact that the same event (Fig. 37) does not occur 
at the same solar time in two places on different meridians of longitude. 
Romer's observations showed that one event seen before a second event at 
one and the same place may really have happened later. The full conse- 
quences of this startling conclusion which has since (p. 330) been estab- 
lished by direct experimental proof in the laboratory is only beginning to 
be grasped. When we look at the moon we are really seeing what it looked 
like one and a half seconds before. Some nebulae are so far away that if our 
telescopes could bring them as near to us as the moon, we should only know 
what they looked like 140 million years ago. One result of Romer's discovery 
was to show how we can calculate the earth's distance from the sun by 
observations on the stars. The calculation based on the assumption that 
the earth moves round the sun like the other planets, agrees with the distance 
calculated from the parallax of a planet, as explained on p. 341, and thus 
established Kepler's doctrine more firmly than ever. 

During the latter half of the seventeenth century the Paris Observatory 
undertook extensive enquiries to adapt and improve the telescope for astro- 
nomical observations. In the latter half of the eighteenth century extensive 
improvements in the technology of glass renewed interest in the study of 
light. Some of the discoveries which resulted have been dealt with in a 



316 Science for the Citizen 

previous chapter. Others which we shall now examine introduce a new feature 
of progress in scientific knowledge. 

So research into the nature of light received a renewed impetus from the 
star lore of the two centuries which witnessed a hitherto unparalleled growth 
of maritime communications and colonization. The way in which enquiry 
progressed was also influenced by the growth of another branch of know- 
ledge. Side by side with the study of light, a different class of physical pheno- 
mena, in which some progress had been achieved in very remote times, 
began to arouse interest. The origins of music are buried in a very remote 
past. For some reason, doubtless inherent in the structure of our organs of 
hearing, a certain sequence of sounds, the octave scale, came into widespread 
use at a very early date. Aside from Ptolemy's experiments on refraction 
and the mechanics of Archimedes, the construction of musical instruments 
provides one of the rare illustrations of exact measurement applied to terres- 
trial phenomena in classical antiquity. Pythagoras (see p. 77, Chapter II) is 
reputed to have discovered the relation between the length of a vibrating 
string and the note emitted, when stretched at constant tension. If the 
length of the vibrating portion is halved the string gives out a note an octave 
higher. If diminished to two-thirds, it gives a note higher by an interval which 
musicians call a fifth* If diminished to three-quarters a note called a. fourth 
higher, and so on. 

The series of numbers called harmonical progressions in text books of 
algebra are a survival from the somewhat mystical significance which the 
Pythagorean brotherhoods attached to this early discovery in experimental 
science. Though there are few extant data concerning the way in which the 
production of sound was studied in ancient civilizations, there is no doubt 
that a keen interest in the improvement of musical instruments had prompted 
a clear understanding of the nature of the stimulus which excites our auditory 
organs. Thus Aristotle, whose views on other departments of physical know- 
ledge are usually worthless, knew that sound is communicated from the 
vibration of the string to our ears by movements of the intervening air, and 
was also familiar with the fact that a vibration occupies double the time when 
the length of a pipe is doubled. From the everyday experience of the interval 
between lightning and thunder-clap, or the influence of winds and echoes 
in an age when people lived more in the open air and buildings were not 
designed for acoustical perfection, there was ample evidence for the fact 
that sounds are reflected from solid surfaces, that they are transmitted through 
the motion of the atmosphere, and that they travel with a finite speed. 

It is not difficult to recognize how a variety of features in the everyday 
life of the middle ages conspired to revive interest in sound. In particular, 
three may be mentioned. The first was a noteworthy improvement in musical 
instruments. The Alexandrian mechanicians Ctesibius and Hero are known 
to have designed a hydraulic organ. A similar model has been recovered from 
Carthage dating about A.D. 200. The daily ritual of the Christian Church 
encouraged the use of instruments suitable for choral accompaniment. From 
the church organ the device of the keyboard was extended to stringed instru- 

* Thus in the scale of C (CDEFGABC), the interval CC, i.e. eight notes including 
both Cs, is an octave. CG is a fifth, CF a fourth. 



The Sailor's World View 



317 



ments in the fifteenth century. Thenceforth the modern piano evolved from 
the clavichord, the virginal, and the spinet. During the sixteenth century the 
manufacture of musical instruments for secular use was developing into an 
important craft. In this milieu Galileo devoted himself to experiments on 
the vibrations of strings. The publication (1636) of Harmonic Universelle y 
a treatise on the mechanical basis of pitch and timbre in stringed instru- 
ments, composed by his ecclesiastical disciple Mersenne, was the signal of 
renewed interest in acoustical phenomena. 

Two other innovations made a new instrument available for investigation 
into the theoretical issues arising out of the technology of stringed instru- 
ments. Hitherto no attempts had been made to find the speed with which 




FIG. 194. WAVE-LIKE MOTION TRACED BY A PENDULUM ON A REVOLVING 
LAMP BLACK SURFACE 

Just as the horizontal or vertical trace of a circular motion is a periodic motion of 
the pendulum type, the trace of the pendulum motion represented as a graph of dis- 
placements and time intervals is a simple wave-like motion such as can be propelled 
along a skipping rope. The periodic time of the wave motion is the interval between 
two crests or two troughs next to one another. In this case it is 2, i.e. the pendulum 
is a "seconds" pendulum (period 2 seconds). 

a sound travels. The introduction of gunpowder into warfare made the 
experience of lightning and thunder claps more tangible. By using the new 
seconds pendulum to time the lag between the flash and the explosion at a 
measured distance from a cannon, Mersenne and Gassendi found the speed of 
sound to be about 1,400 feet per second. More accurate modern determina- 
tions give approximately 1,120 feet at 15 C. Boyle, Flamsteed of the Green- 
wich Observatory, and Halley in England repeated the observations of 
Mersenne and Gassendi, obtaining results in closer agreement with the correct 
figure, as did also Cassini, Huyghens, Picard, and Romer of the Paris Observa- 
tory. Progress was made possible by the fact that the eighteenth century was 
equipped with what the ancients lacked. Musical technology could make use of 
a convenient device for measuring time in short intervals. 
Closely connected with this was another circumstance. Being without the 



318 



Science for the Citizen 



means of measuring short intervals of time, mechanics made little headway 
in antiquity. The dynamical principles which developed in the design of 
the pendulum clock were specially concerned with periodic, that is to say 
wave-like, motions (Figs. 194, 195). A new logical technique drawn from 




FIG. 195. CONNEXION BETWEEN ROTARY AND WAVE MOTION 

We can find a formula for the wave-like trace of the pendulum by a model. The wheel 
rotates at fixed speed. At the start the radius of the wheel from which a weight is 
hung makes an angle 45 to the peg with the horizontal and this angle decreases at the 
rate of 45 per second. That is, the angular velocity of the wheel is 45 or 77/4 
radians per second, if the positive direction of rotation is anticlockwise. After a time t 
(measured in seconds) this radius thus makes an angle 45 (1 t) with the horizontal 
and the weight is a height x t above its mean position (horizontal radius), where 

*t = r sin 45(1 - r) 
More generally, if the radius starts at an initial angle a with angular velocity b, then 

xi = r sin (a + bt) 

a and b are constants which can be measured in degrees or radians. We can make 
things simpler if we use a stop-watch and start it at the moment the radius is horizontal, 
i.e. measure t from the time when a = 0. Then 

xt r sin bt 

If the wheel revolves through 360 in T sees., T is the periodic time (i.e. the time 
between two consecutive crests or two consecutive troughs) and fc, the angle through 
which the wheel revolves in one second, is 360 /T, so 

x t r sin 360r/T 
or measured in radians, 2ir to a revolution 

X{ = r sin 2;rr/T 

If the wheel makes n revolutions a second, T = 1 /TT and 

xg r sin 2irnt 

contact with the mechanical amenities of everyday life supplied the guidance 
which unaided intellectual ingenuity had failed to devise, and successful 
application of the new mechanics to the elastic vibration of strings furnished 
the clue to new discoveries about the nature of light. 



The Sailors World View 



3*9 



THE NATURE OF SOUND 

Nineteenth-century science inherited from the two preceding centuries a 
framework of metaphors drawn from a pre-existing technology. Navigation 
and water power have now ceased to play a prominent part in the daily 
experience of most of us. For that reason the analogy of "wave" or "current'* 
is liable to mystify us unless we firmly grasp the significance they had for an 
earlier generation of scientific workers. To understand the wave theory of light 
it is first essential to understand how sounds are produced and communicated 
to our ears. 

Sounds may be produced by the vibration of a string as with the violin 
and piano; by the vibration of a diaphragm, membrane, or metal plate, as 



Tuning Fork 




Cylinder 

with, 
smoked 
surface 



FIG. 196 

Simple apparatus for measuring frequency of a tuning fork accurately. The tip of a 
pendulum with a complete period of one half second just dips into the mercury making 
contact in the signal circuit every half period when it reaches the vertical position. 

with the drum, the gramophone, the bell or the telephone; by the vibration 
of a column of air as in a tin whistle or organ; or by concussion between 
solids or liquids as when an explosion or splash is heard. Common to all these 
sources of sounds is the fact that air is set in motion to and fro. If air is 
exhausted from a vessel in which an electric bell is suspended no sound is 
heard when the current is turned on. 

That different qualities of sound are produced by different rates of vibration 
is easily shown in various ways, of which the simplest is to fix a hair or thiii 
pointer of tissue paper to one prong of a tuning fork, and trace a wave-like 
image of its movements, when struck, on the smoked surface of a revolving 
cylinder (Fig. 196). If the speed of the cylinder is constant, two tuning forks 
which emit the same musical note will trace out the same number of crests on 
the same length of smoked surface, and tuning forks which emit different 
musical notes will trace out different numbers. The number of wave crests (or 



320 Science for the Citizen 

troughs) traced out in a second is called the frequency of the tuning fork or the 
frequency corresponding to the note. It can be conveniently determined if a 
signal of essentially the same construction as an electric bell, designed to tick 
out seconds, is made to write on the revolving surface below or above the trace 
of the vibrations which the tuning fork executes. What are called pure musical 
notes correspond to a fixed frequency of some vibrating object like the 
prongs of a tuning fork, the air column of an organ pipe, the string of a 
harp, or the diaphragm of a gramophone. The frequency of middle C is 261 
vibrations per second.* Few bodies when struck vibrate as a whole. So most 
sounds are combinations of notes of different frequency, harmonizing or 
otherwise with our aesthetic preferences. Hooke's law of the spring shows 
how any body can be made to execute vibratory or periodic motion when 
stretched; and if you apply the argument on p. 293, you will see that the 
frequency of vibration of a string can be calculated from its length and mass, 
if Hooke's elastic constant k has been previously found. Halving the length 
at fixed tension doubles the frequency and increases the pitch by the interval 
that we call an octave. 

Each vibration of a source of sound sets up an alternating sequence of 
compression and rarefaction in the air immediately in contact, as illustrated 
in Fig. 199. Whenever the air is compressed, the air in its immediate vicinity 
is rarefied. So a train of alternate regions of compression and rarefaction 
spread out in all directions from the vibrating source, just as ripples spread 
out from the spot where a stone strikes the surface of the pond. At any 
position in the wave train the air is alternately compressed and rarefied 
with the same frequency as the vibrator which produces the sound. Thus 
a pure musical note corresponds to a particular frequency of vibration in the 
air between the instrument and the ear. The proof of this lies in the pheno- 
menon of resonance, which is the basis of all reproduction of sound by 
instruments like the gramophone and telephone, or magnification of sounds 
by loud speakers. If a tuning fork is made to vibrate near another of the 
same frequency the latter takes up the note and is itself audible, when held 
close to the ear. This only happens if both tuning forks have the same 
frequency. The only way in which we can easily imagine how this could 
happen is that the air vibrates in unison with the two forks. 

A flat membrane or plate when struck emits a great variety of vibrations, 
and will consequently resonate to a great variety. This fact underlies the 
construction of the gramophone. The instrument used for making the record 
is essentially like the one used for playing it. A flat diaphragm transmits 
vibrations in unison with the frequency emitted by a neighbouring source of 
sound to the holder of a hard steel needle, which makes minute indentations 
on the blank record. When a needle similarly attached to a diaphragm moves 
over these minute notches, it forces the diaphragm to execute similar vibra- 
tions which communicate motion to the surrounding air precisely similar 
to the air vibrations which were impressed on the blank record. The drum 
of the human ear is a resonator of a similar type to the diaphragm of the 
gramophone. It only responds to vibrations between a lower frequency of 
18 per second and an upper frequency in the neighbourhood of 30,000 per 
* New Philharmonic pitch. The old value 256 is used in our numerical illustrations. 



The Sailor's World View 



321 



second. The limits are not quite the same for all people. Some individuals 
can hear the shrill squeak of a bat. Others cannot hear notes of such high 
frequency. The limits also vary very greatly in different species of animals. 
Some animals can guide their movements in the dark, because their ears 
pick up vibrations which our own ears cannot detect. 

The vibration of the air between one tuning fork and another vibrating 
in resonance with it might be imagined to happen in one of two ways. First 
we might suppose a column of air surging to and fro like a piston rod fixed 
between the two prongs, or a column of water forced backwards and forwards 
through a tube. There are several reasons why this cannot be true, one 
being, that sounds do not travel instantaneously. The only alternative is that 




arm B 



I I 




FIG. 197. SIMPLE APPARATUS FOR SHOWING INTERFERENCE OF SOUND 

the vibrations are transmitted through the air in the way described, just 
as ripples on a pond spread out from a region where a stone is thrown or a 
stick is stirred in it. This alternative is supported by two other phenomena 
connected with the production of sound. One is interference. The other is 
the "Doppler effect." 

Generally speaking, the effect of hearing two identical notes played 
together is an increase in the volume of sound. In certain circumstances 
simultaneous notes of the same frequency from two different sources inter- 
fere. That is to say, neither is heard. One way in which the phenomenon 
can be demonstrated is to strike a tuning fork, hold it near the ear, and rotate 
it by the handle. It will then be noticed that in certain positions, at the same 
distance from the ear, the sound is intensified, and in other positions no 
sound is heard. A better demonstration which shows the nature of the 
phenomenon is to blow a whistle in the end of a tube with two arms (Fig. 197) 
leading to the ear. If the length of one arm can be altered by a sliding tube 
like that of the trombone, the sound of the whistle alternately increases to 

L 



322 Science for the Citizen 

a maximum and is silenced as the length of the adjustable tube is graduall 
diminished or made greater. In this device there is only one vibrator. So i 
the air surges to and fro as a single column in each of the two arms leadin 
to the ear, altering the length of one of them would not prevent the ai 
column in it from keeping in step with that in the other. The alternativ 
theory that sound travels in alternating regions of compression and rare 
faction makes it easy to understand what happens. If the lengths of the tw 
arms are not the same, a region of compression at the end of one arm ma; 
coincide with a region of rarefaction at the end of the other, and vice vers 
alternately (Figs. 199 and 200). 

To form a clearer picture of the nature of interference, it is necessar 
to recall that sound travels with a definite speed in a given medium. In ai 
at 15 C. this is about 1,120 feet per second. The time which elapse: 
between seeing the flash and hearing the noise of an explosion is directly 
proportional to the distance. For practical purposes we assume that th< 
transmission of light from the source to the eye is instantaneous, thai 
is to say, the interval is negligible as compared with the time which elapsej 
between the emission of the sound and the moment when we hear it. Tc 
say that sound travels in air at about 1,100 feet per second, therefore, means 
that if we are a mile away from a gun, approximately 4| seconds will inter- 
vene between seeing the flash and hearing the sound, or 9| seconds if we are 
2 miles away. With modern apparatus, the speed of sound in water (about 
4,700 ft. per sec.) can be found by connecting a microphone with an electric 
signal recording on a revolving cylinder, while some delicate time-signal,, 
like a vibrating tuning fork, gives a simultaneous time tracing. The micro- 
phone picks up an echo as well as the original detonation. The wave trace 
of the time marker gives the interval which elapses between the detonation 
and the echo. Thus if the time marker is a tuning fork emitting middle C, 
and 16 wave crests intervene between the two microphone signals, the in- 
terval is 16 ~ 256, or one-sixteenth of a second. In this time (i) the sound 
has reached the sea floor and returned, traversing twice the depth (d). So 
if the depth is known, the speed is easily calculated, being 2 d -~ t. 

Once this speed (s) is known the depth of any ocean can be found, by 
the same method, since d = \st. This is how "soundings" of ocean depths 
are made in modern oceanography. Another application of the speed of sound 
is made in warfare to determine the position of a concealed field battery, at 
an unknown situation X. Three microphones are placed at measured dis- 
tances from one another, and the intervals between the time when the sound 
reaches the first and the time when it reaches the other two is recorded at 
headquarters with electric signals. The reconstruction is shown in Fig. 198. 
The fact that sound travels with a definite speed implies that the path 
along which the disturbance set up by the vibrator proceeds must be divided 
up at any instant in a regular way, as shown in Fig. 199. The string of a 
violin or diaphragm of a telephone vibrates to and fro, like a piston, pressing 
on the neighbouring layers of air and sucking back. Matter does not change 
its state of motion instantaneously. The force with which a stretched string 
or diaphragm presses or pulls on the air in contact with it is measured by the 
acceleration it imparts. This means that it takes time for a column of air to 



The Sailor's World View 



323 



move forward with the same speed as something pressing on it or pulling 
it. So the string or diaphragm can only move by forcing the air in front of 
it to occupy a smaller or larger space. A vibrator such as the string of a violin 
when stretched towards the hearer compresses the layers of air immediately 
in front of it, and this region of high density moves forward with a speed S 
(Fig. 199). If we call the time of a complete vibration T, we may divide the 



Station C 




FIG. 198. SOUND RANGING IN ARTILLERY 

A, B, C are stations with microphones at known distances apart. The microphones 
record simultaneously with an electric signal and timekeepers (on the same general 
principle suggested by Fig. 196) at Headquarters. X is the hidden field battery. The 
sound is heard first at A. At B it is heard t seconds, at C, T seconds after A. If S is 
the speed of sound, it travels St feet during t and ST feet during T seconds. Hence 

XB - XA - St 
XC - XA - ST 

Describe circles about B and C of radii XB XA and XC XA respectively. The 
location of X is now the trigonometrical problem of finding the centre of a circle 
which just grazes the circumference of the first two circles and also passes through A. 

period between a forward swing (1) and the next forward swing (5) into three 
phases. In (2) the string has swung back to its unstretched position, and is 
neither pressing on the neighbouring layers of air nor sucking them back. 
A quarter of a period later (in 3) it has swung backwards as far as it will go, 
sucking on the neighbouring layers of air, and creating a region of low density. 
At the end of the next quarter of a period the string is again unstretched 
and the surrounding air is at normal pressure. At the end of another quarter 
period it has swung forwards as far as it will go, creating a region of high 
pressure in the layers of air just in front of it. During the complete period T 
the first compression has moved over a certain distance from the string, where 



324 



Science for the Citizen 



the neighbouring air layers are also in a state of compression. So the distance 
W between two adjacent regions of compression (or two regions of rarefaction) 
in the track of the sound is the product of the time T and the speed S, or 

W- ST 

If the string is vibrating with a frequency of n complete vibrations (forward 




6 



FIG. 199 

Diagrammatic representation of successive changes of pressure in a sound track, 
starting from a vibrating string, showing how the speed of sound (S) is connected 
with the "wave length" (W) and period (T) of time occupied by one complete 
vibration 

Speed = Distance -r Time 
= W -r T 

If one vibration takes T seconds, -~r vibrations occur in one second. If the frequency 
is n per second 



Hence 



S - 



The Sailor's World View 325 

back to the forward stroke) per second, it makes one vibration in \\n seconds, 



T 

i.e. T = 

n 



Hence, according to the hypothesis we are going to test, the speed of sound, 
the frequency of the vibrator, and the "wave-length" W, are connected by the 
simple formula: 



The reason for calling the distance W a wave-length is that every layer of 
air in the track of a sound suffers compression and rarefaction in alter- 
nate half periods of the vibrator, just as every particle of water in the 
path of an advancing wave alternately moves above and below the normal 
level of the water when at rest. If we could make a graph (Fig. 203) by 
plotting time along the x axis and the pressure of any thin slice of air in 
the sound track above (+) or below ( ) the normal pressure of the air, it 
would be exactly like a graph of the upward and downward movements of a 
cork, floating on water as a wave progressed along the surface, made by 
plotting time along the x axis and the displacement of the cork above (+) 
or below ( ) the resting level of the water along the y axis. It would also 
be mathematically equivalent to the horizontal or vertical displacement 
of a body moving in a circle, as seen in Fig. 195. 

We have seen that the phenomenon of interference is very difficult to 
explain by any hypothesis other than the one which is illustrated in Fig. 199, 
and we can now put it to a more severe test. To be useful it must do more 
than explain how interference might occur. Since a scientific hypothesis is 
first and foremost a guide to conduct, it must tell us how to establish the 
conditions in which interference will occur. For simplicity we will take 
as the source of sound in an experiment like the one in Fig. 197, a tuning 
fork giving 280 complete vibrations per second (just below middle D and 
above C sharp). To make the calculation simple let us also suppose both arms 
A and B in Fig. 197 are 1C) feet long. The wave-length of the note in air is 
given by 

1,120-280 W (feet) 
.*. W 4 feet 

So, to begin with, the sound track in each arm is 2| wave-lengths (Fig. 200). 
Imagine an instant of time when the air in the neighbourhood of a prong 
of the tuning fork may be rarefied. At the point where both arms' lead into 
the earpiece a region of compression will then exist. Half a period later the 
air at the beginning of the sound track will be compressed, and at the earpiece 
it will be rarefied. Suppose that we now extend the arm B to 12 feet, making 
it half a wave-length longer. The disturbance at its end will be a compression 
when the disturbance at the end of A is a rarefaction, and vice versa. In the 
earpiece compressions and rarefactions arrive together, each neutralizing the 
effect of the other. So the air in the earpiece will remain at rest and no sound 
will be heard. If B is now extended to 14 feet (3| W) so that it exceeds the 
length of A by a complete wave-length, the air at the ends of both tubes will 
be in a state of low, normal, or high pressure simultaneously. The sound 



326 



Science for the Citizen 



will be loud again. On extending B by another |W to 16 feet silence will 
result,, and so on. In practice the apparatus shown in Fig. 197 is not the best 
type for putting the hypothesis to the test of actual measurement, since the 
sound track does not follow a straight line. However, it is easy to devise 
apparatus with which we can show with great accuracy that silence occurs 
at any point where the disturbance, arriving along one sound track, differs 
by a half a period, or an odd number of half periods, from the disturbance, 
arriving along another sound track, from the same vibrator. 



Both, tubes same 




Lower tube J \V longer 




Lower tube IV loruyir 




FIG. 200. INTERFERENCE IN Two SOUND TRACKS DIFFERING BY HALF A WAVE-LENGTH 

There are several other ways in which this view of the nature of sound 
can be tested. In the last chapter we found that Hooke's law allows us to 
calculate the frequency with which a weight bobs up and down if we know 
the density, physical dimensions, and elastic constant K, of the spring. Air 
also has the characteristics of a spring, and Hooke himself established an 
analogous law for the "spring of the air" as he called it. If we have found 
the elastic constant for air, we can also calculate the frequency with which 
a column of particular length will vibrate. Hence the frequency of the note 
given out by an organ pipe can be calculated, and so also, as Newton first 
showed, the velocity of sound in air."* As a check on the hypothesis both 
these quantities can be found by other means. 

* The frequency of a vibrating string is given by n = ^T\/ where n is the number 

of vibrations per second, / is the length in cm F is the force or tension in dynes and 
m the mass of the string in gm. per cm. length. 



The Sailor's World View 



327 



We can also infer the existence of a phenomenon which we now encounter 
in everyday life, though the circumstances for verifying it did not exist when 
our present belief about the nature of sound was first tested out. If you have 
been standing by a railway track when an express is blowing its whistle as 
it passes you, you may have noticed a sudden change of pitch at the moment 
of passing. This is an example of what is known as the Doppler effect, after 
the physicist who pointed out its existence, as a necessary consequence of 
the theory, before it had been noticed. Its chief interest lies in its bearing 
on the nature of light. To understand it, we must draw a distinction which 
is not necessary so long as we suppose that the person who hears a sound 




FIG. 201. SEVENTEENTH-CENTURY ILLUSTRATION OF GLASS JAR CONNECTED WITH A 
VACUUM PUMP TO SHOW THAT THE SOUND OF THE BELL is ONLY HEARD WHEN AIR 
is PRESENT IN THE JAR 

is at rest relative to the source of sound, or moving at a slow speed compared 
with the speed of sound in air. To begin with we distinguished musical 
notes by the frequency or period of the vibrator. Thus middle C is the note 
which we hear when a violin string vibrates as a whole, with a frequency of 
256 vibrations per second or a period of aioth of a second. If we are, com- 
paratively speaking, at rest this is transmitted to us along a track in which 
at any given instant successive layers of compression or rarefaction are 
separated by a distance 1,120 4- 256 = 4 feet 4| inches. We might there- 
fore equally well say that the note we are describing is one which reaches us 
by a sound track of wave-length 4 feet 4| inches. Suppose, on the other 
hand, that we are moving rapidly in the direction of a source of sound. The 
time taken for a disturbance, starting at the vibrator, to reach our ears will 
be less or greater, according as we are moving towards or away from it. 
Relatively to ourselves the speed of sound is greater or less than it would be 
if we were at rest. The speed of a sound relative to anybody who hears it is W. 
So if the relative speed of sound changes, either W or n must be different from 



328 Science for the Citizen 

its previous value. We might therefore say that the wave-length remains the 
same, and the vibrator appears to have a different frequency. Since the fre- 
quency of the vibrator is directly measurable, we may equally well prefer to 
say that the wave-length has changed. Whichever we choose, the fact remains 
that the vibrator no longer appears to have the physical characteristic which 
defines the particular note which we hear when we are at rest. 

The argument is more easy to grasp if you consider the person hearing 
the sound to be at rest, and the source like the engine whistle moving towards 
or away from him. If the vibrator in Fig. 199 is at rest along the sound track, 
a compression starting as in (1) covers a distance W during one complete 
period of vibration, by which time another has been generated. In the period 
T the distance covered would be ST feet. This is then the actual wave- 
length of the sound track. If the vibrator is moving in the direction of the 
sound track travelling towards the observer with a speed v, it will move 
forward through vT feet in a time T and the next compression begins 
ST vT feet behind the preceding one. Thus the wave-length will not 
be ST but (S v)T. If the source moves away from the person listening 
to the sound, it will be ST + vT feet from the wave front when the next 
compression is generated, and the wave-length will be (S + z>)T. So at the 
moment when a train passes by a person listening to the whistle, the sound 
experiences a sudden increase of wave-length from (S vJY to (S + z;)T, 
i.e. by an amount 2vT. 

To make this more definite, imagine an aeroplane flying at 191 miles per 
hour or approximately 280 feet per second, close to a high building from 
which a steam whistle is blowing, the pitch of the whistle being two octaves 
above middle C, i.e. a frequency of 1,024. As it approaches, the wave-length 

of the sound is - . -- ft. = 10 in. (approx.). If the aeroplane were 

at rest a wave-length of 10 inches or 5/6 foot would correspond to a frequency 
of approximately 1,344 vibrations per second. A vibrator of this frequency 
would emit a note nearly a third of an octave higher than one with frequency 
1,024 on the natural scale. So the aviator hears a much higher note than the 
man in the street. On passing the building the wave-length of the sound 

which reaches him is " ^ = 1 foot 4 inches (approximately). A 



sound of this wave-length would correspond to the note given out by a 
vibrator of frequency 840. So the aviator hears a note about two-fifths of 
an octave lower than the man in the street. The total change which the 
aviator experiences as he passes over the building is a sudden increase of wave- 
length 2z>T, i.e. 560 -f- 1,024 feet or approximately 6 inches. 



THE NATURE OF LIGHT 



Although the hypotheses which have been advanced in the study of sound 
involve difficult reasoning, they do not come into violent conflict with our 
first-hand impressions of nature. Similar hypotheses, which have been very 
fruitful in leading to the discovery of new knowledge about the nature of 
light, introduce us into a realm of bewildering paradoxes if we start with 



The Sailor's World View 329 

wrong ideas about the nature of scientific enquiry. Science is not a photo- 
graphic picture of the real world which exists independently of our views 
about it, and will continue to exist when we are no longer part of it. It is an 
ordnance map which guides us in finding our way in it. The mountain 
peaks are not painted brown because they really are of that colour. The 
brown is put there to show us where it would be waste of time to build a 
railway track or appropriate to erect a sanatorium. The enduring fact about 
the real world is that there is (at the time of speaking) a mountain peak in 
such and such a place. The colour is the hypothesis which is useless without 
the key provided by the colour scale at the foot of the map. If you grasp 
this firmly you will find no paradox in the statements that light travels in 
waves, and that there is nothing in which the waves travel. 

Much that is written about science for people who want to know more 
about it merely consists of a scaffolding of metaphors of this kind. To say 
that light travels in waves is a metaphor which means as much as the brown 
colour of the mountain peak. Everything depends on whether you know 
the colour scale which tells you the height. Useful and fruitful facts are the 
permanent contribution of scientific enquiry to the edifice of human know- 
ledge, and hypotheses are the scaffolding of metaphors. This may be illus- 
trated by taking a useful and fruitful fact of great antiquity. To say that the 
obliquity of the ecliptic is approximately 23 J makes it possible for anyone 
to calculate his latitude, knowing the day of the year, or to find the day of 
the year if he knows his latitude. Its usefulness as a guide of conduct is not 
diminished in the slightest degree, even if we no longer accept the hypothesis 
which led the Babylonian priesthood or the Chinese astronomers to discover 
it about three thousand years ago. 

Romer's hypothesis, that light travels with a finite speed, followed about 
thirty years after the first determination of the speed of sound by Mersenne, 
who perfected the seconds pendulum. Four of the satellites of Jupiter revolve 
very nearly within the plane of its orbit, and are hidden in its shadow once 
in nearly every revolution. Romer had determined their periods and orbits 
accurately enough to know how much error could arise in calculating at what 
intervals eclipses will recur, and the calculation can be verified whenever the 
Earth and Jupiter are in the same position relative to the sun. On calculating 
the time of an eclipse when Jupiter was near conjunction from the time at 
which a particular eclipse occurred when it was in opposition, he found 
that the observed time was retarded by 16| minutes. 

There are several conceivable ways in which he might have modified his 
original hypothesis. For instance, propinquity to the earth might exert some 
specific influence on the motion of the satellites. Alternatively the explana- 
tion does not lie in their motion, but in the conditions for observing them. 
The second possibility is suggested by our experience of sound. Suppose at 
one place we observe that a gun fires at exactly noon every day. If we set a 
watch accordingly, and move ten miles farther away from the gun, we shall 
not hear it when the dial points to 12 on the following day. The sound will 
reach us about 47 seconds after noon. This retardation is not the only physical 
fact conveyed by saying that sound travels with a certain finite speed. By 
itself it might be due to the fact that the time of firing changes every time 

I* 



330 Science for the Citizen 

we happen to move. This can be rejected for two reasons. One is that the 
same experience also happens to other people. The other is the measurable 
time which elapses between hearing a sound and its echo. 

According to Romer's observation,, an event which was timed for, let us 
say, midnight occurred at 16f minutes after. This retardation, which occurs 
when the distance of Jupiter is increased by the width of the earth's orbit 
(2 x 93,000,000 miles), is one of the physical facts implied by saying that 
light travels at a speed of 

2 x 93,000,000 

-5 = 186,000 miles per second 

Itojf X bO 

However, there is more than this implied in the statement that light 
"travels" at 186,000 miles per second. We can make various calculations 
analogous to those which we have made in connexion with measurements of 
sounds, if we pursue the metaphor suggested by the track of bullets, as 
Newton did, or by a succession of ocean waves, as Huyghens preferred to 
do. Thus astronomical retardation agrees with measurements that can be 
made in the laboratory corresponding to measuring the retardation of an 
echo. It is possible to measure the interval which occurs between the time a 
beam of light leaves its source and is reflected back again. This interval 
depends, like the retardation of a sound echo, on the total path which the 
beam of light traverses. 

A simplified apparatus showing the essential features for measuring the 
ratio of the distance to the retardation, i.e. the speed with which light 
^travels," is seen in Fig. 202. Light falls from the source S on a semi-trans- 
parent mirror at A. Some of it passes through the latter, and some of it is 
reflected towards another mirror C, situated at some considerable distance 
(e.g. about 10 miles). Near A at B there is a toothed wheel between the notches 
of which the light has to pass on its way to C and on its return journey to the 
semi-transparent mirror A, where some of it passes through to be seen by the 
eye. If the wheel is rotated at a low speed the brightness of the beam which 
reaches the eye is first reduced until it eventually disappears when a certain 
speed is reached. Then it reappears and on doubling the speed it becomes 
bright again. A further increase results in dimming and so on. If we compare 
the incident and reflected light to a shower of bullets bouncing back from a 
target along their original path, there must be a speed of rotation at which the 
light passing through the middle of a notch has only just enough time to reach 
the mirror C, and get back to the wheel before the next tooth obstructs its 
path, so that all the light reflected from the first mirror is intercepted at its 
return journey. If n is the speed at which disappearance first occurs, the 
time taken for this to happen, when the wheel has x teeth, as shown in the 
legend of Fig. 202, is 

25 SeCS ' 

During this time the light traverses twice the distance D between B and C, 
so the speed of light is 

2D ~ r = 4nxD 
2nx 



The Sailor's World View 



331 



If the speed is further increased to m revolutions, the wheel rotates from the 
middle of a notch to the middle of the next tooth but one. This is the speed 
at which disappearance next occurs^ the speed of light being 



2D -r 

Of course, m must be approximately three times n, if the reasoning applies 
to the facts, and the two determinations should lead to the same result. 



Whad 




* 



CB is very great compared 



Ey& 



5 




FIG. 202. SIMPLIFIED APPARATUS FOR FINDING THE SPEED OF LIGHT 
If there are x equally spaced teeth and notches (6 in the figure for simplicity) and a 
is the angle between the middle of a notch and the middle of the next tooth. 

360 



i.e, a = 30 if there arc 6 teeth and 6 notches. A wheel turns through 360 in one 
revolution. If it rotates at n revolutions per second, it goes through (360 X ri) in a 
second 3 and through one degree in 

- seconds 



and through a in 



360 



300 x n k 
1 



= - seconds 



2x 360 x n 2nx v 
This is the time taken to rotate between the middle of a notch and the middle of the 
next tooth. To rotate between the middle of a notch and the middle of the next tooth 
but one, it must go through an angle 3a, i.e. 

3 360 , 

- x degrees 

2i X 

If the speed of revolution is then m per second, the time taken to do this is 

3 

- sees. 



332 Science for the Citizen 

Actually it is best to take the average of the two. For instance, suppose the 
wheel B and the mirror C are 5 miles apart, the number of teeth 720 and 
the speeds of rotation at which disappearance occurs first and next are 13 and 
38 per second. We have now two values for the speed of light 

4 x 13 x 720 x 5 = 187,200 miles per second 
t X 38 x 720 x 5 = 182,400 miles per second 

The mean in this case would be 184,800 miles per second. Terrestrial 
determinations of the speed of light agree with the astronomical estimate 
with an accuracy of one part in ten thousand. 

According to the reasoning used in the design of this experiment and the 
calculation of results, the situations in which changes in the real world 
excite our organs of hearing and vision resemble one another in three ways. 
Using the customary abstract noun as a convenient metaphor, we say three 
things about light. First, light follows a definite path. The space between 
a source of light and the eye has local characteristics. If it is charged with 
dust particles a "beam" is visible to a second observer, and shadows are 
only cast where objects are placed in the region limited by the beam. Second, 
a beam of light can be turned back on its own course, or as we say "reflected," 
like the echo of a sound. Third, the time at which an object appears to be 
visible depends on how far it is separated from the observer's eye. 

But for one unfortunate circumstance there would be nothing remarkable 
about these similarities. Hearing is only possible if there is a substantial 
connexion, e.g. air or water, between the organ of hearing and the source of 
sound, or between any resonator which acts as a source of sound and another 
vibrator to which it responds (Fig. 201). This is not true of light. We can 
see objects separated from our eyes by space which is not filled by anything 
which we can weigh. If, for instance, an electric light bulb and an electric 
bell are suspended in a bell jar from which all the air is exhausted, the bell 
becomes inaudible till air is readmitted, while the light is just as visible 
throughout the experiment. This is a formidable objection to pursuing the 
analogy further. When we say that sound travels in waves, we mean two 
things. One is that at any region in the sound track, the air or other medium 
becomes successively more and less dense than it would otherwise be. The 
other is that at any instant of time a measurable distance separates successive 
places where the medium is more or less dense. The first is what we mean by 
frequency. The second is what we mean by wave-length, and the two are 
connected with the speed of sound in a definite way. Since light can pass 
where there is nothing weighable, its path cannot be mapped out in regions 
of varying density in a literal sense. Consequently the similarity seems to 
break down. 

In spite of this there are a variety of facts about light which encourage 
us to look for further similarities, and we may be less surprised to find this 
is so if we bear in mind the real nature of the similarity between a sound 
wave and a wave of the sea. The fundamental similarity is that the displace- 
ment of a particle of water in the track of a sea wave and the density of a thin 
layer of air in the sound track can be represented graphically in the same 



The Sailor's World View 



333 



form (Fig. 203). Measurements which involve the way these quantities vary 
from time to time can, therefore^ be expressed algebraically in the same form. 
The resemblance between two patches coloured dark brown on an ordnance 
map does not lie in the physical property of colour. Mountain peaks are 




FIG. 203 

The graphical representation on the variations of density in a sound track is a wave 

form. 

generally white. The key to the resemblance, implied in saying that sound 
is a form of wave motion, lies in the sort of mathematics which is used in 
connexion with sound measurements. 
A phenomenon which encourages us to search for similarity between the 




FIG. 204 

Obstruction of light by two crystals of Tourmaline placed with their long axes at right 

angles. 

kind of measurements which we can make in studying light on the one 
hand and sea waves on the other, is a change which sometimes occurs when 
a beam of light passes through a mineral crystal. This phenomenon called 
polarisation was first studied by Huyghens. It is very well seen when a 
beam passes through a crystal of tourmaline cut into two slices parallel to 
its long axis. If the slices are placed in line so that the long axes are 
parallel (Fig. 204) a source of light can be seen through them., though not 



334 



Science for the Citizen 



as brightly as when the crystals are taken away. There is no visible change 
if both halves are rotated simultaneously. If one is rotated slightly while the 
other is held in position, the source becomes dim. When the long axes of 
the two crystals are at right angles complete darkness results. Rotation through 
another right angle restores the original brightness. The crystal acts like a 
grating. It only allows some of the light to pass through, and the part of 
the beam which does so can pass through a second crystal in which, to 
use the same metaphor, the slits of the grating are parallel to those of the 
first. Though a hailstorm of bullets might seem a more appropriate parable 
for the experience that light passes through a vacuum, the analogy of 
waves is the only one which fits polarization (Fig. 205). Bullets could pass 
freely wherever the slits of the grating crossed. Waves sent along a skipping 
rope held between the slits of a grating can only pass if they occur in the 





FIG. 205. WAVES LIKE DISPLACEMENTS ALONG A SKIPPING ROPE OBSTRUCTED BY 
Two SLITS AT RIGHT ANGLES 



plane of the slits, and are completely obstructed by a second grating with 
slits at right angles to those of the first. 

The wave analogy was suggested by Huyghens, the bullet analogy by 
Newton. Newton's reputation was in the ascendant, and further theoretical 
interest in optical theory declined after Newton's work on the spectrum, till 
practical scientific instrument makers solved the problem of making achro- 
matic lenses. The issue of Dollond's patent in the latter half of the eighteenth 
century prompted a commercial demand for large telescope lenses of high- 
grade glass, and the renewal of interest in the theory of light followed the 
invention of a process for making optical glass in thick homogeneous slabs 
during the closing years of the same century. There was steady progress 
during the first three decades of the nineteenth century, an enormous im- 
provement in optical instruments, and with it a new drive to theoretical 
research. For example, the Committee of the Royal Astronomical Society 
commissioned Faraday in 1824 to investigate the chemistry of optical glass 
production; and Foucault, the French physicist, invented new methods for 
polishing large discs of flint glass. In England Young, and in France, where 
Laplace was the centre of an influential school of astronomers, Foucault and 
Fresnel, returned to the problem which had been neglected for more than a 



The Sailor's World View 

century, and elucidated the phenomena of interference and diffraction 
which had emerged already in Newton's work on colour fringes. 

You may say that Huyghens' hypothesis was merely reasoning from 
analogy. In our childhood and adolescence we are continually warned against 
the fallacies that beset this method of argument The truth is that analogy 
is a very powerful instrument of scientific reasoning, and most of the really 
fruitful known facts about nature have been discovered by reasoning from 
analogy, often analogy of a very crude kind. Analogy like any other instru- 
ment in the technique of reasoning is helpful or harmful according as it is 
used to suggest further enquiry or to close the door to it. If, like a man of 
science or a craftsman, you want to get something done, you cannot afford 
to run the risks of inexperience. So analogy will point the road to new dis- 
coveries. If, like an economist, you want excuses for leaving things as they 
are, analogy will provide you with as many as you want. 

Among the class of fruitful facts which the wave analogy has stimulated 
people to discover, the phenomenon called interference provides the simplest 



FIG. 206. INTERFERENCE BANDS FORMED BY Two MIRRORS ON A SCREEN 

illustration of the way in which the same kind of mathematics can be used 
in calculating measurements of light or waves. We have seen in what cir- 
cumstances two sounds can combine to produce no audible effect. In 
analogous circumstances two beams of light can produce darkness. To 
demonstrate interference of sound directly we have to use sound tracks 
of the same wave-length, and this is generally done by forcing sound to 
reach the air by paths of different length from the same vibrator. The 
white light by which we usually see things is not pure. It is a mixture (p. 169) 
of several kinds of coloured lights, just as most of the sounds we hear are 
the result of simultaneous vibrations of different frequency. If two notes 
of a chord were sounded together in a tube like the one shown in Fig. 197 
we should not get complete silence as the result of changing the length of 
one arm. An extension of the adjustable tube sufficient to produce inter- 
ference of one note would not lead to extinction of the other. Increasing its 
length would result in hearing first one note alone, next the other note alone, 
then the complete chord, and so on. Many situations occur in which we see 
a region of brightness split up into bands corresponding to the complete 
chord of the spectrum alternating with coloured fringes like the spectra 
produced by prisms. You have probably seen this effect when the moon is 



336 



Science for the Citizen, 



surrounded by clouds, and you can always observe it by looking at the sun 
between the gap of two fingers pressed close together. In the laboratory it can 
be produced in various ways, such as focussing light on a screen from a narrow 
slit through two edges of a prism simultaneously, or by reflection from two 
mirrors inclined at a very small angle. Another way is to make light pass 
through a very fine grating of parallel slits, like the fine slits between the hairs 
of your fingers. When pure spectral light, like the yellow glow produced when 
a pinch of common salt is thrown in the non-luminous flame of an oil cooker 
or blow pipe flame, is used instead of white light, bands of pure coloured light 
alternate with streaks of complete darkness. 



Obtuse 





FIG. 207. Two ARRANGEMENTS FOR SHOWING INTERFERENCE BANDS 

The distance which separates these interference bands has a definite 
relation to the conditions of the experiment. It varies according to the dis- 
tance of the source and screen from the prism, etching, or mirrors, the angle 
of inclination of the mirrors or prism faces, and the distance apart of the 
etchings in a "diffraction" grating. If the same conditions are reproduced the 
bands are the same distance apart so long as the same kind of pure light 
is used. So we can associate a particular measurement with light of a particular 
colour, just as Pythagoras found that a particular length of a string is charac- 
teristic of a particular musical note. By itself, the discovery of Pythagoras 
could not tell Ctesibius how long to make the pipes of his organ. So likewise, 
by themselves, the measurements which tell us how to get interference bands 
for a certain kind of light with two mirrors do not tell us how to get the same 
result with a prism. Before it was possible to calculate the dimensions of an 



The Sailor's World View 337 

organ pipe, which will give a note equivalent to one produced by a string 
of known length, thickness, tension, and elasticity, it was necessary to find a 
number, which defines it independently of the nature of the instrument used 
to produce it. This number is its wave-length. 




FIG. 208. CALCULATING THE DISTANCE APART OF TLM Two IMAGES FROM THE SAME 
SOURCE WHEN Two MIRRORS ARE INCLINED To ONE ANOTHER AT AN ANGLE 180 u -fc 3 

b BEING SMALL 

The image of the slit S in the upper mirror appears to be at A, the same perpendicular 
distance (AX.) behind the mirror as separates the mirror from the slit. The ray so 
striking the mirror at its edge O also appears to come from A at the same distance 
irom O as S, i.e. OA = OS. The image of S in the lower mirror appears to be at B, 
and for the same reasons SY ^ BY and OB =- OS. Hence OB = OA and the two 
images A and B, together with S the slit 3 lie on the same circle of radius OS = r. 

The triangles ZOY and ZXS are both right-angled with a common angle c. So the 
third angles b are equivalent The angle ASB ( = &) stands on the same arc as AOB, 
and since the angle which an arc subtends at the centre (O) is twice the angle it 
subtends at any point (S) on the circumference, AOB = 2b. If b is measured in 
degrees;, the length of the arc AB is 

2-nr X 2b 4irrb 
360 r ~360~ 

If b is very small the arc AB differs by a negligible quantity from the chord AB 
which is the actual distance of one image from the other. 

So although a wave is merely a metaphor, a wave-length, if we care to 
use the same expression, is not. The number which enables us to make 
measurements on the interference of light embodies a physical truth about 
the way in which we succeed in making correct measurements of changes 
which occur in the real world. By pressing the analogy of wave motion 
farther we obtain a set of numbers which have a relation to different regions 
of the spectrum similar to the relation between the wave-lengths of notes in a 



338 Science for the Citizen 

musical chord. A wave-length in sound is a number applied to a distance. 
One of its characteristics is that if the distance traversed by two sound tracks 
simultaneously reaching the ear from the same vibrator differs by half the 
distance called the wave-length of the note it gives out, no sound is heard. 
Clearly, there is no reason why we should not find a corresponding distance 
connecting the length of two beams of light which reach the eye from the same 
source. With two mirrors this can be done in the way shown in Figs. 207 
and 208. 

The mirror experiment depends on placing the source of single-coloured 
light, which is a thin slit (S), in such a position that the paths of any two rays 
reflected on the screen from the two mirrors are of unequal length. Light from 
a plane mirror appears to come from a point equidistant behind it. The paths 
of any two rays which meet at a point on the screen where they reinforce 
or interfere are equivalent to the distance of the point from the images 
(A and B) of the slit behind the two mirrors. If the mirrors are tilted towards 
one another through a very small angle b (i.e. the angle between them is 
180 &), so that the distance of their common edge from the slit is r, you 
will see from Fig. 208 that the distance between the images (d) is related 
to their distance (r) from the source of light by the simple formula 

4777* 
d = -^-7-7 (when b is measured in degrees) 



So to get the distance between the two images it is only necessary to measure 
the angle at which the mirrors are inclined and the distance of their common 
edge from the slit. At any point P or Q along the length of the illuminated 
patch where the two reflected beams overlap on the screen, the paths of the 
two reflected rays differ by a small distance which is calculable. If the differ- 
ence at P is p and the difference at Q, separated from P by the distance PQ, 
is q, Fig. 209 shows that if the mid-point between the two images is separated 
from the screen by a distance D 



If the screen is separated from the common edges of the two mirrors by a 
distance s, and is perpendicular, or nearly so, to the plane bisecting the angle 
between the mirrors, D = s + r. According to what we have agreed to mean 
by a wave-length, if P is a point in the middle of a dark interference band, the 
difference in the length of the paths traversed by the interfering rays is either 
| a wave-length, or H wave-lengths, or 2 wave-lengths and so on. For sim- 
plicity suppose it is 31W. Interference will next occur if the difference of 
the paths is 21 or 4-iW. So if Q is the middle of the nearest interference band 
separated from P by the distance PQ, q is either 2 J W or 4| W. Whatever p is, 
the difference between p and q is always W, i.e. 



or _-_ 

r W "~ 360(5 + r) 



The Sailor's World View 



339 



The distance (PQ) between the middle of adjacent streaks of darkness can 
be measured accurately with a microscope. It is then found that the value of 
W for yellow sodium light (from common salt) is 0-0005893 millimetre 
(0-0000232 inch). 



Source A 



Source, B^_._ y^ _____ 



%d 



*'M 
P^ I y 

^ f'^J 

Q^- '---4- 



FIG. 209. DIFFERENCE BETWEEN THE LENGTH OF Two RAYS IMPINGING ON THE 
SAME POINT (P OR Q) OF A SCREEN EQUIDISTANT [FROM Two SOURCES OF LIGHT 

A AND B 

The point P is separated by a distance x from O, the point vertically opposite the mid- 
point of the line joining A to B. The distance of the sources from the screen is D and 
from one another d. By Pythagoras' theorem 

AP 2 -- (!d + x) 2 H D 2 

- -]~d 2 + dx + x* + D a 
BP 2 = (Arf x) 2 -\ D 2 

- Id* dx + x* + D a 
.*. AP 2 BP 2 =- 2dx 

:. (AP - BP) (AP + BP) - 2dx 

If P is near O, and D is great compared with d, AP -f- BP does not differ appreciably 
from 2D. 



- BP - - 
B1 ~ 2D ~ I? 



If Q is another point and OQ y 



AQ - BQ - 



dy 
D 



If p is the difference in length of the rays meeting at P (i.e. AP BP), and q is the 
difference in length of the rays meeting at Q (i.e. AQ BQ) 

d f , 

g P =".g(y x) 



This measurement, like the wave-length of a musical note, is independent 
of the instrument. By applying the law of refraction in the same way as we 
have here applied the law of reflection we can calculate the position (d and D) 
of the two images when yellow sodium light (from common salt) is passed 



340 Science for the Citizen 

simultaneously through two faces of a prism. The same result is obtained 
from the measurement of the interference bands, i.e. W = PQ(d D). 
With pure lights of different colour different values of W are found. For 
describing wave-lengths in whole numbers it is now usual to express them 
in 10-millionths of a millimetre (called an angstrom). Each colour which the 
eye recognizes as distinct corresponds to a certain range of wave-lengths, 
extending between roughly 4,000 and 8,000 angstroms. The middle of the 
violet region of the spectrum corresponds to about 4,200 A. ; the middle of 
the blue to about 4^500 A.; the middle of the green to about 5,200 A.; the 
middle of the yellow to about 5,800 A.; and the middle of the red to about 
7,000 A. 

Although the mathematics of wave motion is not particularly easy, it has 
one comparatively simple application which brings out the fruitfulness of the 
wave metaphor as an aid to measurement in a very spectacular way. This is 
the fact that we can calculate from measurements based on a phenomenon 
comparable to the Doppler effect, a value for the earth's distance from the 
sun in close agreement with the value obtained from the parallax of a near 
planet (p. 210). The spectrum of sunlight is crossed by fine dark lines. In the 
spectra of light from some of the stars similar lines appear. Measurement of 
where these lie by photographing their spectra side by side with that of a 
terrestrial source of light (like sodium light or the iron arc) shows that they 
undergo a slight shift in the course of the year, according as the earth is 
moving in its orbit towards or away from the particular star whose spectrum is 
studied. The pitch of sound is different when we are moving towards or away 
from the source. The shift in the wave-length of the dark lines in starlight is 
analogous. If the speed of the observer (v) is known, the shift can be calculated 
from the speed of light (S) by the Doppler formula 2z>T, in which T is the 
period of the vibrator (p. 328). If the wave-length is W when the observer is at 
rest, T = W ~ S. So that the shift from a wave-length W x when the observer 
is moving towards the source to W 2 when he is moving away from it is 
2z>W -f- S. The wave-length of the sound heard by an observer at rest may be 
taken as the mean of the wave-lengths (W 1 and W 2 ) of the sound when the 
observer is in motion. An analogous calculation may be made for the annual 
shift in the spectra of the stars. 

For a star suitably chosen the observer's speed (v) can thus be the rate at 
which the earth rotates in its orbit. If r miles is the mean distance of the earth 
from the sun, the circumference of the orbit is 2vrr miles, which it traverses 
in 365 days, or 365 x 24 x 3,600 seconds. Twice in the course of a year, at 
intervals of six months, a physicist examining the spectrum of such a star can 
do so when it is moving directly in the line of sight either towards or away 
from the source, with a speed 

v = 27rr~ (365 x 24 x 3,600) miles per second 

Having found the wave-lengths of various sources of pure light, it is possible 
to graduate the whole spectrum in wave-lengths. Hence the shift of a line (or 
the middle of a broad one) can be expressed as a wave-length. If we sub- 
stitute the value of the shift, the speed of light (S), and the wave-length corres- 
ponding to the mean position of the band in Doppler' s formula for sound. 



The Sailor*s World View 



341 



we can get a value for z>; and from this r is found to be approximately 92| 
million miles, which agrees to less than one part in two hundred with the 
parallax value. The Doppler effect can also be used to calculate the rate at 
which the fixed stars appear to be moving away from us. A readable account 
of recent speculations on the expanding universe is given in the 1935 edition 
of Whitaker's Almanack and in J. G. Crowther's Progress of Science. 

Researches of this kind may seem to be very remote from the everyday 
life of mankind. This is far from true. The initial stimulus which spectroscopic 
research received from the progress of glass technology, after the long period 
of inertia subsequent to Newton's work, was reinforced by the discovery 
of chemical regularities in spectra at a time when chemical manufacture 
was actively encouraging exploration into new fields of enquiry. How spectro- 
scopy has helped us to build dirigibles and make the coloured lights of adver- 
tising signs will be explained later to be one of the crowning victories of man's 
conquest of materials. How the extension of the wave metaphor led to the 
discovery of wireless transmission which has broken down so many social 
barriers of space and time will appear in the story of man's conquest of 
Power. The distinction between fruitful and useful facts which was made 
elsewhere is more superficial than real. Information that is fruitful in acting 
as a check on methods of discovery and a correct hypothesis leads in the 
long run to the kind of knowledge which can be used to provide more of 
the means of life and leisure. 



Tftc lrner 6caZc 



3 


: 4 

tenths ; 










: i 








| 


\ 




7 -"> J' ' 




=9div 1 V OIL upper scale. 



FIG. 210. THE VERNIER SCALE 

Invented by Pierre Vernier, a French mathematician, early in the sixteenth century, 
the vernier scale is an auxiliary movable ruler device, which permits great accuracy. 
On the lower ruler a distance equivalent to 9 divisions on the upper scale is divided 
into 10 parts. To measure an object the end of which is marked by the thin line between 
3 2 and 3 3 in the figure, set the beginning of the vernier scale at this level and look 
for the first division on the vernier which exactly coincides with a division on the 
upper scale. In the figure this is the second division and the correct measurement is 
3-22. The theory of the device is as follows. If x is some fraction of a division on the 
upper scale to be ascertained, the correct measurement is 3-2 -j- x. The first whole 
number a of the smaller divisions on the lower scale which coincides with the upper 
differs from a divisions on the upper scale by the distance x. Now 10 divisions on the 
lower scale is i ' (T of a division on the upper. Hence 

(^ >fl ) + x = a 

:. x = &a 

If a is 2, x 2 tenths of a scale division on the upper scale. If a division on the upper 
scale is 0-1, x =- 0-02. 



342 



Science for the Citizen 



THE "ABERRATION" OF LIGHT 

During the period which immediately followed Newton's work on colour 
fringes the only important advance in the study of light had been independent 
confirmation of Romer's view from an unexpected source by Bradley, the 
Astronomer Royal, in 1728. The confidence which Newton's theory conferred 
on the Copernican doctrine stimulated renewed attempts to obtain decisive 
evidence of the earth's orbital motion from the detection of a measurable 
displacement of the position of a fixed star at times when the earth is at opposite 
ends of a diameter of its orbit. In the early part of the seventeenth century 
accuracy of measurement had been greatly advanced by two inventions, the 
telescope and the vernier device shown in Fig. 210. If the earth revolves around 
the sun, there should be detectable differences in the R.A. and declination of 
any star at different seasons. The maximum displacement of R.A. (see Fig. 211) 
occurs when the star's R.A. differs from that of the sun by 6 hours or 18 hours. 
There is then no difference in declination. The maximum difference of 
declination should occur (Fig. 212) when the sun's R.A. differs from that of 
the star by hours or 12 hours. There should then be no parallax in R.A. 
From these seasonal differences we can calculate the star's heliocentric parallax, 
which is approximately the ratio of the sun's distance to the star's distance 
from the earth. 

These parallaxes are very small. For the nearest star (Fig. 213) of great 
brightness visible from the latitude of London Sirius the angle of parallax 
is only about one-third of a second of arc (0-37 1")., i.e. one ten thousandth 
of a degree; and there were no instruments sufficiently delicate to measure 
such small differences until the beginning of the nineteenth century, when 
the new patent of Dollond, improved manufacture of optical glass, Foucault's 
method for polishing lenses, and the theoretical researches into achromatism 




R.A.1SH. 



Mar. 21'Sun enters T 

FIG. 211 

When the sun's R.A. differs from the star's by 6 hours or 18 hours, the parallax is 
in R.A. only, and there is no parallax in declination. If the star's distance from the 
solar system is d y and the radius of the earth's orbit r, tan p=^r~dord=r- tan />, 
The star chosen is y Draconis with R.A. 18 hours. It is not actually in the plane of 
the earth's orbit, but considerably north of it. 



The Sailor's World View 



343 



which accompanied the progress of scientific glass technology, bore fruit in 
great improvements of telescope construction and in the invention of the 
compound microscope. 



Pole of % 
'Z~-Y Draconis 




,,/>v x A^ t23 
<>"-T-PX->- 




21 



FIG. 212. RRADLEY'S OBSERVATIONS ON y DRACONIS 

When the sun's R.A. is the same as the star's (or differs by 12 hours), there is maximum 
parallax in declination and no parallax in R.A. The star drawn is y Draconis, which is 
taken to be near the pole of the ecliptic: it is actually about 15 away. Remember 
(see Fig. 127) in reading this figure that the earth is "in Capricorn" when the sun is 
"in Cancer" and vice versa. The R.A. of y Draconis is approximately 18 hours, i.e. 
its lower transit is at midnight about December 2 1st, and its upper transit at midnight 
June 21st. On these dates its decimation should exceed its mean value D (the 
declination of the star from the celestial equator with the sun as its centre) by the 
minute parallactic angle p or -| p. This is now known to be 0-17, an angle 
too small for Bradley to measure. On the contrary he found that the declination, d 
on March 2 1st, when there should be no parallax in decimation, was less than D, and 
that the declination J 2 on September 23rd, when there should also be no parallax 
in declination, was greater than D by one-third of a minute of arc (20 -5"). 
On March 2 1st the earth is travelling towards the direction from which the angle D 
is measured, i.e. it is moving away from the celestial pole. On September 23rd it is 
travelling away from the direction from which the angle D is measured, i.e. it is 
moving towards the celestial pole. 

Among many other contemporaries of Newton, Bradley was engaged in 
the search for a detectable star parallax. He chose a star in the constellation of 
Draco. The star y Draconis lies about 15 from the pole of the ecliptic. For 
simplicity it is drawn as if it were at the pole of the ecliptic in Figs. 212 and 
216, which more nearly describe the position of w Draconis, only 3 from the 



344 



Science for the Citizen 



ecliptic pole. The R.A. of y Draconis is very nearly 18 hours. So it should 
show maximum displacement of declination about June 21st and December 
21st. Bradley could detect none with his instruments. On the contrary he 





BC 
FIG. 213. SINES AND TANGENTS OF VERY SMALL ANGLES 

The sine of the angle a is BD ~ AD and the tangent is BD AB. When a is very 
small, the arc CD does not differ appreciably from the perpendicular BD, and the 
base AB does not differ appreciably from the radius of the circle AD = r. So sin a 
= CD -.- r and tan a = CD r. If the angle a is measured in degrees the arc CD 
is a(277r) 4- 360. If it is measured in seconds of arc, it is 2-nra ~ (360 x 60 X 60). 
Hence sin a" and tan a" are each 

2zra 44<z 



360 x 60 x 60 
So if p" is the parallax of a star 

tan/>" 



7 X 36 2 X 10 s 



7 x 36 s x 10 8 
star's distance = sun's distance ~ 



7 x 36 2 x 10 3 
__ 93,000,000 X 36 2 X I0 3 v 7 
~~ 44/> 

- 192 X 1Qn 

P 
The parallax of Sirius is 0-371". Hence the distance of Sirius is 

192 X 10 14 -7- 371 = 52 x 10 12 or 52 billion miles. 

found a displacement which was greatest about March 21st and September 
23rd, the dates on which the declination should not be affected by parallax 
at all. Bradley confined his observations to the measurement of declination. 
The difference in declination of two stars is simply the difference of the two 



The Sailor's World View 



345 



zenith distances at transit (D = z.d. 4- lat.) if measured at the same latitude. 
An equivalent difference in R.A. is also found when the star's R.A. is the same 
as the sun's or differs from it by 12 hours, i.e. when there should be no 
difference due to parallax. 

According to Cajori the clue to this anomalous behaviour came to him from 
a class of everyday experiences which greatly helped to elucidate the laws of 
the combination of motions in different directions, when navigation was at 
the mercy of wind and current. "Accompanying a pleasure party on a sail on 
the Thames one day about September 1728," he noticed that the wind seemed 
to shift each time "that the boat put about, and a question put to the boatman 
brought the (to him) significant reply that the changes in direction of the vane 



r..fl_j 

L & 




2 /^(2^. 



Iscc. 



FIG. 214. THE TILT OF THE TELESCOPE 

To envisage the significance of Bradley's observations, imagine a train of bullets from 
a machine gun fired at the mouth of a cylinder with its opening pointed directly towards 
the oncoming bullets. If the cylinder is moved with a speed d ft. per sec. in a direc- 
tion at right angles to the movement of the bullets., the latter will not pass straight 
down the middle of the cylinder. If the motion of the cylinder is sufficiently swift they 
will strike against the sides. To prevent this the cylinder must be tilted in its direction 
of motion. At a suitable tilt the bullets will pass straight along the middle of the 
cylinder. The propagation of a wave front of light along Bradley's telescope was analo- 
gous. That is to say, the object glass of the telescope has to be tilted slightly towards 
the direction of the observer's motion. 

at the top of the mast were merely due to change in the boat's course, the wind 
remaining steady throughout. ..." The significant fact about the shift called 
"aberration" which Bradley first observed was that the greatest displacement 
of the star's position is found at times when the earth is moving in its orbit 
either towards or away from the direction from which the angle is measured 
(Fig. 212). This fact receives a simple explanation from the fact that light 
travels with a finite speed. Hence the direction from which light reaches an 
observer is the resultant of his motion and that of the light. If the light is to 
pass down the centre of the telescope and be seen by the eye, the object-glass 
of the telescope must be tilted slightly in the direction of the observer's motion 
(Fig. 214). If the telescope is used to measure the angle between the star's 
direction and some direction of reference (e.g. a direction along the celestial 
equator in Fig. 212, or along the ecliptic plane in Fig. 215), then the angle is in- 
creased if the observer's motion is away from the direction of reference and 



346 



Science for the Citizen 





FIG. 215. ABERRATION OF LIGHT 

If the observer is at rest with his eye at B, light strikes the objective of his telescope 
at A, and traverses the distance AB in x seconds. The angle D is the elevation 
of the star above the plane of the ecliptic in which the earth is moving, i.e. the 
celestial latitude of the star. This is the true direction of the star. The observer 
is moving away from the source of light in the upper figure and towards it in the 
lower figure, in the line BC, and traverses the distance BC in x seconds. If he is 
moving away from the source and holds his telescope at the angle D, his eye 
reaches C when the light has traversed the distance AB. So he does not see the source. 
To see it, the telescope must be tilted upwards, i.e. in the direction of his motion, at an 
angle d 2 so that the light strikes the objective at A when he is at D before he gets 
to B, and does not reach B till he and his telescope have traversed the distance DB, 
x seconds later. Hence DB ~ BC. If he is moving towards the source of light as in the 
lower figure the telescope must be tilted downward, i.e. in the direction of his motions, 
making an angle J x . Since light passes over a distance AB in x seconds, and the speed 
of light is AB #, the observer's speed is BC -f- x 

AB ^ Speed of light 
BC Speed of observer 

In the upper figure the exterior angle d% = D + a 2 . In the lower figure the exterior 
angle D d l -f a p i.e. d l D a*, 



The Sailor's World View 



347 



decreased if his motion is towards this direction. For example, in Fig. 212, 
declination is measured from a line on the celestial equator plane, joining the 
sun to the R.A. circle of the star. On March 21st the earth is moving in this 
direction, and since the telescope must be tilted towards the direction of motion, 
the angle D is decreased. On September 23rd the earth is moving in the 




= distance iray- 
eZIccZ by observer 
in umt time 



Observers Observer's 

modem nuotiun 
Sept. 23 March 21 

FIG. 216 

Since y Draconis is fairly near the pole of the ecliptic, its celestial latitude, L, may 
be taken as about 90. But on March 21st, tilting the telescope in the direction of 
the observer's motion decreases the observed latitude by a small angle, a, to / x . 
Similarly on September 23rd, the angle is increased by a to / 2 . 

Now tan a ~r^s and since a is a very small angle (see Fig. 213) 

= BD 

eartn?s orbital speed 
speed of light 

opposite direction, and tilting the telescope towards the direction of motion 
increases the angle. The plane of the earth's orbital motion is not the 
celestial equator, but that of the ecliptic. So the elevation of the star from the 
ecliptic, i.e. its celestial latitude (see p. 220) measures the inclination of the star- 
beam to the observer's motion. This can be calculated from its declination and 
R.A. by an application of the formulae in the appendix to Chapter IV. When 



348 Science for the Citizen 

the celestial latitude is 90, i.e. when the star is at the pole of the ecliptic, the 
correct set of the telescope simply depends on the speed of the earth in its orbit 
and the speed of light, just as the correct angle for aiming at a pheasant flying 
directly overhead (p. 239) simply depends on the speed of the bird and the 
speed of the bullet. If r is the distance of the sun from the earth, the observer 
is moving through 2-rrr = 4 7 4 r miles in 365 days. His speed is therefore 

44r llr 

: miles per second 



7 x 365 X 24 x 60 x 60 42 x 365 X 36 X 10 2 

If the star is at the pole of the ecliptic, the angle a is 20 -5", i.e. 20-5 ~ (60 
x 60) degrees, and: 

speed of earth's orbital motion 

tan a = , ,.- 

speed of light 

If the speed of light is 186,000 miles per second, 

llr 



tan a = 



42 x 365 x 36 X 10 5 x 186 



The tangent of this angle can be found without recourse to tables (Fig. 213), 
being 

20-5 77 41 x 22 

6(Tx 60 X 180 == 72 x 18 x 7 x~To 3 
So we may put: 

llr 41 x 22 



186 X 42 x~365 X 36 X 10 5 72 x 18 X 7 X ~1$ 
__ 41 x 22 x 42 x 186 X 365 X 36 X 10 2 __ 9259 X 10 () 
.-. r = 11 x 72 x 18"x~7 ' ~~~ 998~"x "To 2 approx ' 

To three significant figures this is 92,800,000 miles, in close agreement with 
the sun's distance calculated from parallax or from the annual shift of the spectra 
of the fixed stars. 



THE SURVIVAL OF WORD MAGIC 

During the nineteenth century, when the phenomena of interference were 
first fully investigated, the wave metaphor became firmly entrenched in 
scientific discussion. It was the fashion to speak of a supposititious all- 
pervading ether, which remained when all weighable matter had been 
removed from a space. The plain truth is that this ether was less a description 
of what we encounter when investigating the world than a description of the 
kind of mathematics with which we calculate what happens in certain 
circumstances. In the study of how chemical qualities of substances (like a 
pinch of common salt) are related to the spectra which they yield when 
made to glow in a non-luminous flame, recent discoveries have received less 
help from the wave metaphor and more from the analogy of the bullet. 
Consequently the ether is receding into the realm of word magic along with 
phlogiston (p. 422), the life force and the Real Presence. 

Science has nothing to lose from the decease of a metaphor. In a different 
social context from the one in which we live, the change might be regarded 
as a welcome release from reliance on the existence of something beyond 



The Sailor's World View 349 

the range of our senses. If we wish to understand why the new outlook in 
physics is interpreted as a reason for rejecting the materialistic temper of the 
century which began with Diderot and ended with Darwin, we must not expect 
to find an answer in the progress of scientific knowledge. The story of man's 
conquest of Substance and of Power will make it clear that we have reached a 
situation in which social machinery can no longer accommodate rapid advance 
in discovering the material amenities which science confers. We are thus 
faced with two alternatives. One is to devise the social machinery which will 
ensure the further progress of science by finding social uses for new dis- 
coveries. The other is to discredit confidence in the fact-loving method of 
science. This can be done most effectively by surrounding the work of the 
scientific investigator with an air of mystery, so that he fulfils the role of the 
first men of science in human history. The ancient calendar priests were the 
first scientists and the first civil servants. They ceased to be scientists as they 
aspired to become masters. The world tires of its masters and finds a growing 
need for efficient civil servants. So we may hope that the adventure of human 
knowledge will survive all the efforts to reinstate mysticism in the present 
period of cultural decay. 

The phase of discovery which has been dealt with in this chapter draws 
attention to aspects of scientific method often overlooked, especially in the 
study of social institutions. There are two errors commonly held among people 
who study human society. The one to which the economist is prone is the use 
of facts to illustrate hypotheses based on seemingly self-evident principles, 
instead of the use of facts to test whether seemingly self-evident principles are 
a safe guide to conduct. At the opposite extreme is the school of historians who 
eschew all hypothesis and discourage "rash" generalization, apparently 
content to collect facts as an end in themselves. The truth is that scientific 
knowledge is not a mere collection of facts. It is an organized repository of 
useful and fruitful facts. Some of the most useful and fruitful facts have been 
found by using theories which seem utterly absurd in retrospect. What 
specially distinguishes the method of genuine science is that theory and 
practice, hypothesis and fact, work hand in hand. In a scientific experi- 
ment the investigator sets out to collect the facts that he expects to get. 
If they do not exist he fails to get them. If he fails to get them there must be 
something wrong with his expectations. So if he is a good investigator he will 
either discard his hypothesis or modify it and submit it to further test in its 
modified form. A scientific hypothesis must live dangerously or die of inani- 
tion. Science thrives on daring generalizations. There is nothing particularly 
scientific about excessive caution. Cautious explorers do not cross the Atlantic 
of truth. 

THINGS TO MEMORIZE 

1. Wave-length x frequency = speed of sound 1,120 feet per second 
at 15 C. 

1 /F 

2. Frequency of vibrating string n = ^\ 

21 \ m 

3. The change in wave-length observed when someone moving with a 
velocity v passes a source of frequency T is 



350 Science for the Citizen 

EXAMPLES ON CHAPTER VI 

Take the velocity of sound as 1,120 feet per second, except where another value 
is indicated. 

1. What is the wave-length of a note if the velocity of the waves is 1,100 feet 
per second and the frequency is 440? 

2. If the shortest wave-length a man can hear is about 18 cm. and the 
longest about 900 cm. calculate the frequency in both cases. Take the velocity 
of sound to be 33,000 cm. per second. 

3. What is the wave-length of a pure note if the speed of propagation is 
340 metres per second and its frequency is 256 per second? 

4. A steamer nearing the coast blows a whistle and after 10 seconds an echo 
is heard. When the whistle is blown five minutes later the echo is heard after 
8 seconds. Calculate how far from the coast the steamer was when the second 
whistle was sounded, and her speed. 

5. How far away is a thunderstorm, if a flash of lightning is seen 4 seconds 
before a clap of thunder is heard? 

6. A man standing in a ravine with parallel sides fires a rifle. He hears an 
echo after 1 J seconds and another 1 second later. After another 1^ seconds he 
hears a third echo. How do you account for these echoes, and how wide is the 
ravine? 

7. The echo of the blast of a ship's siren is heard in 1J seconds. If the 
reflection is caused by an iceberg, how many seconds after hearing the echo 
will it be before the ship, steaming at IS-^j- miles per hour, hits it? 

8. An engine, blowing a whistle whose frequency is 500, passes a man in a 
signal box at a mile a minute. What does the frequency of the whistle appear to 
be to the man, before and after the engine passes? 

9. When half a mile away from a tunnel through a hill, an engine blows its 
whistle, and an echo is heard by the driver 4 J seconds later. Taking the velocity 
of sound as 1,100 feet per second, how fast is the engine going? 

10. A train rushes past a stationary engine, which is blowing a whistle of 
frequency 600, at 60 miles per hour. What value would the passengers in the 
train give to the frequency of the whistle before and after passing? 

11. A bullet is heard to strike a target 2| seconds after it has been fired 
with a velocity of 2^000 feet per second. How far away is the target from the 
marksman? 

12. A tuning fork has a bristle attached to one of its prongs which just 
touches a vertical smoked glass plate. When the tuning fork is sounded and the 
plate allowed to fall under the action of gravity a wavy line is traced on the 
plate. If a distance d marks off N waves show that the frequency is NV2d~/g. 

13. Using the same apparatus as in Example 12, the plate is dropped from 
rest through a distance of 1-8 cm. In the next 10-2 cm. fall the fork makes 
35 waves. What is the frequency of the fork? 

14. What is the depth of a well, if a stone is heard to strike the water 2 seconds 
after it is dropped? (Take g = 32 ft. sec. 2 ) 

15. When two vibrating bodies have not exactly the same frequency, there 
are times when the compressions or rarefactions from both sources pass a given 
spot simultaneously, and times when a compression from one neutralizes a 
rarefaction from the other. These alternate periods of much disturbance and 
then comparatively little disturbance produce a pulsating effect called beats. 
If two strings of frequencies 300 and 302 per second are bowed, how many 
beats are produced in a second? 



The Sailor's World View 351 

16. Two tuning forks., one of frequency 256 per second, are sounded simul- 
taneously and produce 4 beats per second. When the other fork of unknown 
frequency is loaded with a small piece of wax the beats stop. Determine the 
frequency of this fork. 

17. Two tuning forks of frequency 512 are taken. One of the forks has its 
prongs filed, so that when sounded with the other 5 beats per second are 
produced. Find the frequency of the filed fork. 

18. A noise is heard at three stations A, B and C. A hears the noise 13-14 
seconds before B and 8-89 seconds before C. If the rectangular co-ordinates 
of A, B, and C, are respectively (4,0), (0,0) and (0,2) on a map, a mile being the 
unit, by means of a geometrical construction find the co-ordinates of the source 
of the noise. Take the speed of sound as 1,100 ft. sec. 

19. From Fig. 197 we see that sound waves from a can reach the ear by two 
separate paths. When one path is 16 cm. longer than the other no sound is 
heard. When it is 32 cm. longer a sound can be heard, and when 48 cm. longer 
no sound, and so on. Taking the velocity of sound to be 33,200 cm. per second, 
what is the frequency of the whistle? 

20. As an engine, blowing its whistle, passes a station, the pitch of 
the whistle appears to a man on the platform to drop a minor third, i.e. to 
ths of its apparent frequency before passing. How fast is the train going? 
Take the speed of sound as 1,100 ft. sec. 

21. If the greatest interval between successive eclipses of Jupiter's second 
moon is 42 hours 28 minutes 56 seconds, and the least interval is 42 hours 
28 minutes 28 seconds, determine the speed of light (miles per second), given 
that the radius of the earth's orbit is 92-7 x 10 6 miles. 

22. Using Fig. 202 to determine the velocity of light, find the angular 
velocity in radians per second of the wheel for the third and fourth disappear- 
ance of the image, when the distance from C to B is 10 km. and the wheel has 
720 teeth. 

23. Using the same figure, let the distance from C to B be 15 km. and the 
wheel have 600 teeth. Given the speed of light ~ 3-0 x 10 10 cm. per second, 
how many revolutions in a second must the toothed wheel make so that light 
passing through a gap on one journey is stopped by the next tooth on the 
return journey? 

24. A beam of light on a revolving mirror is reflected by it and travels a 
distance of 750 metres to a mirror, which reflects the beam back along its 
path. The beam is then reflected again by the revolving mirror, and is found 
to make an angle of one minute with its original path. What is the angular 
velocity of the revolving mirror in revolutions per second? 

SELECTED REFERENCES FOR PART I 

As introductory text books on the subject matter of Chapters III, V, and VI, 
the following are specially recommended; 

Jeans' Mechanics. 

Houstoun's Intermediate Physics. 

Hadley's Everyday Physics. 

and for more advanced reading on the subject matter of Chapter IV : 

Smart's Spherical Astronomy. 
Ball's Spherical Astronomy. 



352 Science for the Citizen 

The following are useful sources for material in connexion with the social 
background of Part I : 



ANTHIAUME, l/ABB^: U Evolution et FEnseignement de la Science Nautique en 

France. (Paris: Dumont.) 1920. 
ANTHIAUME, L'ABBE: Uhistoire de la Science Nautique anterieurement a la 

Dccouverte du Nouveau Monde. (Le Havre.) 1913. 

BENSAUDE, JOAQUIM : UAstronomie Nautique au Portugal a VEpoque des Grandes 
Decouvertes. (Bern, Akademische Buchhandlung von Max Drechsel). 1912. 
BRUNEI, p.: Histoire des Sciences, Antiquite. (Paris: Payot.) 1935. 
BURGEL, BRUNO H. : Aus Ferncn Welten. (Berlin: Verlag Ullstein.) 1920. 
DOBELL, CLIFFORD: Antony van Leeuwenhoek and His Little Animals. (Bale.) 

1932. 
FERRAND, GABRIEL: Introduction a V Astronomic Nautique Ardbe. (Paris: Paul 

Geuthner.) 1928. 
GALILEO GALILEI: Dialogues Concerning Two New Sciences. (New York: The 

Macmillan Co.) 1914. 
HAKLUYT, RICHARD: The Principal Navigations, Voyages, Traffiques and Dis- 

coveries of the English Nation. (Text of 1589.) (Dent.) 1928. 
HART, IVOR B.: The Mechanical Investigations of Leonardo da Vinci. (Chapman 

and Hall.) 1925. 

HEATH, SIR T. : Aristarchus of Samos. (S.P.C.K.) 1921. 
HOOKE, ROBERT : The Diary of Robert Hooke. Editd. by H. W. Robinson and 

W. Adams. (Taylor and Francis.) 1935. 
HOOKE, s. H.: New Year's Day. (Gerald Howe.) 1927. 
HUXLEY, T. H., AND GREGORY, RICHARD: Physiography. (Macmillan.) 1924. 
LOCKYER, SIR J. NORMAN: The Dawn of Astronomy. (Cassell.) 1894. 
MARGUET, F.: Histoire de la longitude a la mer au XV I IF siecle en France. 

(Paris, Chalomell.) 1917. 

MEYER, E.: Geschichte des Altertums. (Stuttgart and Berlin: J. G. Cotta.) 1928. 
MEYER, E. : Die altere Chronologic Babyloniens, Assyriens undAegyptiens. (Stuttgart 

and Berlin: J. G. Cotta.) 1925. 
MORLEY, s. G.: Introduction to the Study of Maya Hieroglyphs. (Bureau of Am. 

Ethnol. Bull. 57.) 1915. 
NEUBURGER, ALBERT: The Technical Arts and Sciences of the Ancients. 

Methuen.) 1930. 

NEUGEBAUER, o. : Vorgrieschische Mathematik, Bd. I. (Bd. XLIII of "Die 
Grundlehren der mathematischen Wissenchaften in Einzeldarstel- 
lungen," edit, by R. Courant. Berlin, Julius Springer.) 1934. 
NILSSON, MARTIN P.: Primitive Time Reckoning. (Lund: Sweden: C. W. K. 

Gleerup.) 1920. 

PARSONS, L. M.: Everyday Science. (Macmillan.) 1929. 
PETRIE, SIR FLINDERS: History of Egypt, vol. i. (Methuen.) 1923. 
REINHARDT, K.: Poseidonius. (Munich, Beck.) 1921. 
REINHARDT, K. : Kosmos und Sympathie. (Munich, Beck.) 1 925. 
REY, ABEL: La Science Oriental Avant les Grecs. (Paris: La Renaissance du 

Livre.) 1930. 
SCHMIDT, w. (edit.): Hero Alexandrinus. Bibliothec. script. Grace. Rom. 

(Teubner.) 1899. 

SMITH, DAVID EUGENE: History of Mathematics (2 vols.) (Boston, Ginn.) 1923. 
SNYDER, CARL: The World Machine. (Longmans.) 1907. 

SPRAT, THOMAS: The History of the Royal Society of London* for the Improving of 
Natural Knowledge. (London.) 1722. 



The Sailor's World View 353 

TAYLOR, E. G. R.: "Jean Rotz: His Neglected Treatise on Nautical Science/' 

The Geographical Journal* May 1929. 
TAYLOR, E. G. R. : "A Sixteenth-Century MS. Navigating Manual in the Society's 

Library/' The Geographical Journal,, October 1931. 
TAYLOR, E. G. R.: Tudor Geography, 1485-1583. (Methuen.) 1930. 
THOMPSON, j. E.: The Civilisation of the Mayas. (Field Museum, Chicago, 

Anthrop. Handbook 25.) 1927. 
THOMPSON, j. E.: A Correlation of Maya and European Calendars. (Field 

Museum, Chicago, Anthrop. Series vol. 17, 1.) 1927. 
THORNDIKE, LYNN: History of Magic and Experimental Science. (2 vols.) 

(Macmillan.) 1934. 

THULIN, C.* Corpus Agrim. Rom. vol. i, 1911. 
TURNER, D. M.: The Book of Scientific Discovery. (Harrap.) 1933. 
WARD, F. A. B. : Time Measurement. Part I. Historical Review Handbooks of the 

Science Museum. (Stat. Office.) 1936. 
WILDE, E.: Geschichte der Optik. Berlin. 1843. 
WOLF, A. : A History of Science, Philosophy and Technology in the Sixteenth and 

Seventeenth Centuries. (George Allen & Unwin.) 1935. 
WRIGHT, J. K.: Notes on the Knowledge of Latitude and Longitude in the Middle 

Ages. (Isis. vol. v, pt. i, Brussels.) 1923. 
WRIGHT, J. K.: The Geographical Lore of the Time of the Crusades. (American 

Geog. Soc.) 1925. 



M 



PART II 

The Conquest of Substitutes 



But the mortallest enemy unto knowledge, and that which hath 
done the greatest execution upon truth, hath been a peremptory 
adhesion unto authority; and more especially, the establishing 
of our belief upon the dictates of antiquity. For (as every capacity 
may observe) most men, of ages present, so superstitiously do 
look upon ages past, that the authorities of the one exceed the 
reasons of the other. Whose persons indeed far removed from 
our times, their works, which seldom with us pass uncontrolled, 
either by contemporaries, or immediate successors, are now 
become out of the distance of envies; and, the farther removed 
from present times, are conceived to approach the nearer unto 
truth itself. Now hereby methinks we manifestly delude our- 
selves, and widely walk out of the track of truth. SIR THOMAS 
BROWNE'S Pseudodoxia Epidemica. 



CHAPTER VII 

THE THIRD STATE OF MATTER 

The Freeborn Miner 

ONE characteristic which specially distinguishes human beings from other 
animals is their power to change the nature of their environment. By selecting 
other organisms for their associates and collecting different sorts of lifeless 
matter from their surroundings, they provide themselves with the means 
of food, of shelter, of locomotion, and of adornment. Mankind has thus 
risen superior to the barriers of climate and situation limiting the distribution 
of other living creatures. In establishing himself as a species with a world- 
wide distribution, man's power to change his environment is circumscribed 
by two facts. A comparatively small number of plants and animals are suit- 
able for cultivation or domestication, and a comparatively small number of 
substances which are found in nature are directly suitable for fabricating 
articles of use or amusement. Organisms and their products suitable to human 
requirements of one or the other sort myrrh and cedarwood, silk and 
pearls had a very limited distribution in the ancient world. So likewise had 
the metals which man learned to treasure first, and later to work. In the ages 
of scarcity, when the trade routes were of fundamental importance to the 
diffusion of culture, man had to discover how to find his way about a world 
in which the basic means of human satisfaction were sparse. Until the end 
of the eighteenth century of our era man's greatest intellectual achievements 
were associated with the survey of a world in which the good things of life 
were very unequally distributed. 

The story of how man learned to find his way about the world has been 
told in the preceding chapters. This stage in the growth of man's under- 
standing of the universe reached a climax in the three centuries which 
witnessed the opening up of the resources of two new continents to European 
civilization. By the end of the eighteenth century the habitable world had 
been explored with the aid of the sextant and telescope, the theodolite and 
chronometer. An inventory of man's resources for continuing a mode of 
living which had changed very little since the construction of the Pyramids 
had been accomplished. So at this point we may fittingly take leave of the 
story of man's conquest of time reckoning and earth measurement. When 
the nineteenth century dawned progress in the knowledge of nature had 
assumed a different aspect. The ages of scarcity were drawing to a close. 
Space and Time were making way for the study of Matter, Power, and Health. 

If we consider the variety of substances available for human use and the 
resources of power which can now replace human effort, the gap separating 
us from the seventeenth century of our own era is far greater than that 
which separates the seventeenth century of the present era from the Mediter- 
ranean world of 1700 B.C. In the long period which intervened there had 
*een no radical additions to knowledge of the use of materials, nor any 



358 Science for the Citizen 

radical improvement in mechanical substitutes for human labour. The 
knowledge of metals, of glass making, of pottery, of weaving, of dyeing, oi 
brewing, of irrigation, of enamels, and of cosmetics, had scarcely surpassed 
the achievements of the early civilizations in the Mediterranean. The plough, 
the wind sail, and the water-wheel, were older than the hanging gardens oi 
Babylon. The best silks were still the silks of Old Cathay. What material 
progress had occurred in the world as a whole was little more than a process 
of give and take. There had been no important new inventions, nor new 
industries, other than those to which reference has been made already 
That is to say, gunpowder, printing, spectacles, and mechanical clocks. 

Our task will now be to trace the story of man's conquest of materials, and 
to indicate the social prospect which it unfolds. What we call the science oi 
chemistry today is made up of rules for making and recognizing the presence 
of substances which have very definite physical properties, such as density, 
colour, crystalline form, melting and boiling points, odour, texture, and sc 
forth. So in textbooks of chemistry it is customary to begin with the definition 
of a pure substance. Perhaps it is better not to do so. There is no way ol 
capturing the meaning of a pure substance in a simple sentence. Any defin- 
ition which we attempt to frame registers the stage we have reached in dis- 
covering reliable recipes for making things. Primitive man collected materials 
from rocks and springs, from animal tissues and plant juices, and used them 
to make metals, pigments and fabrics, perfumes, dyes, and medicines. His 
success and the quality of his products depended on local peculiarities ol 
the minerals, or on local species of plants and animals. All the early industries 
of civilization had the same local character. Sand was mixed with lime 01 
potashes and various minerals found here and there. The quality of the glass 
of a particular locality became famous. The secretion of a Mediterranean 
sea-slug furnished the Tyrian purple of Phoenician trade. It was nobody's 
business to know why the glass of a particular place was better, or to make 
the dye which the sea-slug excreted. Man took the fruits of nature as the} 
came. What science he had was mainly used to get to the places where the} 
were to be found. 

In the new phase on which we are now entering the reverse is true. Ever} 
advance of science makes man less dependent on local materials. We kno\* 
the nature of the dyes and drugs which were once obtained exclusively frorr 
animals and plants, and we manufacture them from substances like coal 
which are more widely distributed. If we cared to do so, we could make 
them from substances which are universally distributed. We no longer depenc 
on the manure dumps of India for the nitrates used for making gunpowdei 
in medieval times, nor on the natural deposits of Chile for the nitrates whicl: 
were introduced for use as fertilizers in the nineteenth century. Nitrates car 
be made from the nitrogen of the air and the common salt of the ocean. Fo] 
metals of great hardness and strength we are no longer dependent on loca 
ores. We know innumerable ways of making alloys. Aluminium, which is th< 
most abundant and universally distributed of all the metals in the superficia 
layers of the earth's crust, is already beginning to displace the use of th( 

ViMVifM* m^talc wliirVi natmnc Viavi* ctnicrcrlp>r1 tr* mrmrmnliv** in tV/ r*<acf- 



The Third State of Matter 359 

still predominate, the basic ingredients of modern industry are of a totally 
different kind from those used in the time of Newton. Modern chemistry 
recognizes more than a quarter of a million distinct substances. Hundreds 
are being discovered yearly, and an enormous number are already employed 
in making articles of everyday use. Three hundred years ago chemical 
industry as we now know it did not exist. Crude lime, made by cooking 
chalk and limestone in kilns, was sold for making cement and glass, or for 
curing leather. Incinerated charcoal or "potashes" rich in alkaline potassium 
carbonate from the forests of central and eastern Europe was imported into 
Britain for cleaning wool fibres and making soap, prepared by boiling the 
solution with lard or mutton fat. Crude alum from the Isle of Wight was 
employed in preparing cloth for dyeing. At most a dozen vegetable juices in 
place of the many hundreds of modern synthetic pigments were in everyday 
use. Salt sold for preserving meat owed its chief commercial use to its principal 
impurity the hygroscopic magnesium chloride, which is removed from the 
best table salt of today. All the pure metals then known gold, silver, copper, 
tin, iron, lead, zinc, mercury, and antimony had been worked in the ancient 
world, and were prepared from their ores by the same processes. Mineral 
pigments like the vermilion cinnabar (mercury sulphide) and green malachite 
(copper carbonate), or cosmetics like galena (lead sulphide), were probably 
used less in sixteenth-century England than in ancient Egypt. More 
medicinal chemicals were recognized, but these had not become important 
articles of commerce. Jewels, which owe their special characteristics to crystal 
form, one of the fundamental criteria of a pure substance in the modern 
sense, represented almost the only articles of commerce which the chemist 
of today would recognize as such. 

To be sure, it would be an exaggeration to say that there was no science of 
chemistry before the nineteenth century, or that no progress was made in 
its study between the age of Archimedes and the age of Newton. No sharp 
line can be drawn between organized scientific knowledge and mere rule-of- 
thumb methods of recognizing the characteristics of nature. The latter 
must always precede the former, and the former grows out of the latter. What 
can be truthfully said is that there was no substantial improvement in the 
actual technique of preparing materials of reliable quality, that is to say 
making "pure" substances, between the fall of the Alexandrian civilization 
and the middle of the seventeenth century of our era. No substantial improve- 
ment in processes of preparing, detecting, and assaying metals were made in 
the first sixteen centuries of the present era, and if Arab medicine had added 
a number of new substances to the list of those previously known, it had 
done little to clarify any general rules for making them. 

The search for pure substances was mainly prompted by three principal 
requirements of everyday life in civilized communities metals, munitions 
(pp. 406-8), and medicines. The enormous influence of the latter may seem 
strange when we reflect upon the scope of reliable medical knowledge as late 
as the time of Jenner. Still, it would be a mistake to regard primitive medi- 
cine as mere superstition. One of the commonest complaints of everyday life 
is constipation, and the possibility of relieving it by the use of various 
reagents was a discovery of great antiquity. The purgative calomel (a 



360 Science for the Citizen 

chloride of mercury) was known to the Egyptians and if the achievements of; 
the physician were less conspicuous in other fields, the gratitude of their 
fellows for the use of aperients and purgatives is amply illustrated by the 
lyrical flourish with which discoveries of new ones were announced. As late 
as A.D. 1648 Glauber could proclaim the addition of sodium sulphate to the 
pharmacopoeia with a tract entitled Miraculo Mundi> and Glauber's salt 
enjoyed the sobriquet sal mirabile. A little later Nehemiah Grew described 
the uses of Epsom salts (magnesium sulphate) with the evangelical fervour 
of an all too familiar advertisement. 

From the practice of mining, ancient chemistry had learned the processes 
now called oxidation and reduction. The history of man's earliest attempts to 
extract metals from their ores is still largely speculative. Both gold and silver 
occur as metals in nature, and both could be shaped with the stone hammers 
which primitive man possessed. Native gold and native silver were treasured 
before the dawn of city life, and were first used for making simple ornaments. 
Metallic copper also exists in the regions where civilization began, and native 
copper seems to have been the source of the first hammered copper instru- 
ments. These are found along with chisels and adzes of gold, silver, and 
electrum (a silver-gold alloy), in the recent excavations at Ur. Little progress 
in the use of metals could be made when only gold and silver were known. 
Though native copper is soft, it is hardened by hammering. Rickard (Man 
and Metals, Vol. I) says : 

As soon as primitive man began to shape his finds of copper he must have 
observed this important fact, which was decisive in making the metal more 
serviceable for the fabrication of instruments. At once the red stone became more 
useful than the yellow. . . . The use of native copper marks the beginnings of 
every ancient metal culture. . . . Perhaps two millennia separated the first use 
of hammered copper from the beginning of true metal culture, when copper 
was smelted from its ores and cast in a mould. 

Elliott Smith has put forward reasons to support a plausible hypothesis 
to account for this momentous discovery. The green copper carbonate called 
malachite is readily converted by a dull red heat into copper oxide, which is 
reduced to the metal itself when roasted with charcoal. Over a long period of 
pre-history there is ample evidence that mankind used powdered malachite 
as a pigment. In pre-dynastic Egypt it was used for facial adornment. When- 
ever debris containing malachite was dropped among the embers of a charcoal 
fire, spangles of metallic copper would be formed. We may well suppose that 
this was not a very rare occurrence. The use of bronze (a copper-tin alloy) 
came much later about 1500 B.C., and its discovery was probably due to 
the fact that ores of copper and tin often occur side by side. Though iron 
occurs with great rarity as a truly indigenous metal apart from its ores, the 
Jovian thunderbolt had familiarized man with it from early times. Fragments 
of meteoric iron are found with human remains long before the age of iron 
instruments began about 1200 B.C., and modern Esquimaux have been 
observed to make knives by inserting flakes of meteoric iron in grooved strips 
of walrus bone. Several common pigments, the ochres such as haematite, are 
rich in iron. So the discovery of smelting iron from its ores may have been 
made by analogy with the pre-existing technique for extracting copper 



The Third State of Matter 361 

Primitive metallurgy began with the reduction of the class of ores now called 
metallic carbonates or oxides, i.e. by heating them with charcoal in a closed 
space. This was all that was necessary for preparing the tin used in making 
bronze from its common ores. Many of the common ores are what we now 
know to be sulphides, i.e. combinations of a metal with the element sulphur. 
The next step in the progress of metallurgy was probably the discovery that 
some of the "sulphureous" ores, like those of lead, copper, iron, zinc, silver, 
and mercury, can be reduced if previously heated in air, or as we now say, if 
they are first "oxidized" The discovery that the process is facilitated by keep- 
ing the air in motion probably led to the invention of the bellows and the first 
crude blast furnaces for oxidizing the more recalcitrant sulphureous ores like 
pyrites (iron sulphide). Natural ores are always mixed with fragments of 
sand, which is mainly composed of the same substance (silica or silicon 
oxide) as quartz. The trick of heating the ore with some "flux" such as lime, 
to make it readily form a molten mass, may have been gleaned from the 
discovery of glass, or it may have led to it. 

These processes were practised more than three thousand years ago, and 
their nature remained an enigma till the end of the eighteenth century. The 
craft of the goldsmith and the silversmith offered great opportunities for 
chicanery, as a familiar legend about Archimedes reminds us. The artificers 
of Alexandria were apparently familiar with various devices for detecting and 
manufacturing fraudulent articles by dissolving metals in acids and reprecipi- 
tating them from their salts. According to a legend current in the alchemical 
works of the Middle Ages, Diocletian ordered the suppression of all the 
Alexandrian books on the art of chemistry for fear that the artificers would 
enrich themselves exorbitantly. An unduly sanguine interpretation of their 
achievements led to a futile search for the supposedly lost secret of trans- 
muting lead into gold. Slowly Europe recovered the real secrets of Alexandrian 
chemistry, without recognizing them as such. 

The technique of preparing pure chemicals for medicinal use owes much 
to the very ancient industry of brewing. The formation of tartaric acid 
crystals in wine vats was already known to the Egyptians. This was possibly 
the origin of the preparation of pure substances by allowing them to crystallize 
out by cooling., and also of the practice of recognizing them by their crystalline 
form. Most important of all devices for separating substances with individual 
characteristics was the process of distillation., which is used in the preparation 
of alcoholic liquors. The retort had already become an important instrument 
of medical research in Alexandria. In the hands of the Arabs, to whom we 
owe the word alcohol,, it became the means of adding many new members 
to the known list of chemical species. The Moorish physicians made some 
advance towards classifying the nature of substances. They recognized 
solutions as acids, alkalis, or of salts, according to their effect on vegetable 
dyes, used in the preparation of fabrics. One of the dyes which was formerly 
used in this way is the familiar "litmus," which turns red when dipped in 
a mineral acid, or blue if dipped in the solution of an alkali (i.e. "bases" like 
sodium hydroxide and quicklime, or "carbonates" of sodium and potassium). 
The Arab chemists gave recipes for making the three chief mineral acids 
(nitric, sulphuric, and hydrochloric) of modern commerce by distilling off 
the vapours formed when various salts are heated. 

M* 



362 



Science for the Citizen 



The nature of distillation, like the nature of reduction and oxidation, 
remained an enigma. In the processes of the laboratory substances seemingly 
disappeared on heating. They became "spirits," and the art of chemistry 
was to make these spirits materialize. The spirits of the retort, which still 
remain in our vocabulary as spirits of wine, spirits of salt, and the like, are now 
recognized to be a third state of matter. In the ancient world what we now 
call gases and vapours were not recognized as a form of matter, and it was 
utterly impossible to see any common thread running through the familiar 
processes of distillation in medicine and reduction or oxidation in metallurgy. 
Although Greek materialism had proclaimed the robust doctrine on which 
modern chemistry rests, neither the experience of the physician nor the art 
of the metalworker seemed to support the belief that matter is indestructible. 
The ancient world recognized two classes of things which could be weighed 



Spirits' 




FIG. 217 

According to the ancient conception of chemical processes the fire of the furnace 
dissolved the matter contained in the retort into its earthy constituent and the "spirits" 
which escaped. That the spirits^ or as we now say gases, had weight was not recognized. 

with scales solids and liquids. Aristotle's physics had rejected the possi- 
bility that air could have weight. 

Before the middle of the sixteenth century there was no clear evidence 
that air is a form of matter in this sense, nor was there the least apprehension 
of the fact that innumerable forms of matter exist like air in the gaseous 
state. So the distinction which we now draw between the two classes of pure 
substances called elements and compounds completely eluded the science 
of antiquity. What was neither liquid or solid was spirit. The discovery that 
air has weight, separating the theory of chemistry today from the practice 
of chemistry till the time of Galileo, Hooke, and Boyle, is one of the great 
dichotomies in the history of human knowledge. Circumstances of everyday 
life contributed in various ways to this discovery, which revolutionized man's 
command over the use of materials. In the great age of the sailing ship men 
of science were beginning to measure forces as matter in motion. This raised 
the question, Is the force of the wind also a manifestation of matter in motion? 
Galileo's experiments on the way in which heavy bodies fall prompted enquiry 
into the material nature of air for another and far more important reason, 



The Third State of Matter 363 

related to the immediate social context of his researches in a different way. 
Laboratory experience showed that heavy bodies gain speed at the same 
fixed rate when falling through air. This seems (see p. 366) to be in flat 
contradiction to experience of everyday life. We shall now see how Galileo 
resolved the paradox, and how his pupils and successors proved the truth of 
his solution. 

The source from which Galileo got his clue made the study of air and its 
characteristics an issue of momentous importance about the same time, and it 
was of special significance in Britain, where the revival of the teachings of the 
Greek atomists was received by men of science with enthusiasm. Extensive 
development of deep-shaft mining took place in the sixteenth and seventeenth 
centuries owing, among other things, to the wastage of iron in warfare after 
the introduction of artillery. Speaking of the change in Britain in a recent 
memoir (Econ. Hist. Rev., 1934), Nef remarks : 

Though the output of tin, unlike other metallic ores, was not increasing, 
even in tin mines technical problems assumed an entirely new importance, 
because the more easily accessible supplies of ore had been largely exhausted 
in the Middle Ages. During the reigns of Elizabeth and her two Stuart successors 
money was poured out lavishly in the construction of hundreds of adits and 
ventilation shafts and of hundreds of drainage engines driven by water. . , . 
The high cost of fuel began to check the expansion of the output of iron before 
the end of Elizabeth's reign. . . . During the reigns of Elizabeth, Charles I, 
and James I, coal was successfully substituted for wood fuel in calcining ores 
. . . and in nearly all finishing processes. 

The mere technique of deep-shaft mining was not a new development in 
human industry. Neuberger's book The Technical Art and Sciences of the 
Ancients tells us : 

Mining was at different stages of advancement among the various ancient 
peoples. It was particularly developed among the Egyptians, who probably 
opened up copper mines on the peninsula of Sinai as early as the third millen- 
nium B.C. Besides these the vast quarries of Turra near Cairo have also been 
preserved; they prove to us that at that very early date open working had 
been given up in favour of shafts. The ancient Egyptians were thus not satisfied 
with merely removing the stones in the hill from the outside, but penetrated 
far into the interior. Wells of the same period, for example, Joseph's well at 
Cairo, descend up to 300 feet vertically into the earth. In view of the fact that 
these shafts were constructed about 2500 B.C., it can hardly be doubted that 
similar ones were also dug out for mining purposes in some cases. The high 
standard of mining construction attained by the Egyptians is rivalled by the 
Indians and the Chinese, who likewise sank pits about five thousand years ago. 

Though the technique was not essentially new, there was an important 
difference between the conditions of deep-shaft mining in the seventeenth 
century and the practice of more ancient times. Mining in antiquity was made 
possible by abundant supplies of slave labour, which was used with what 
now seems to be almost an incredibly reckless disregard for human life. 
Neuberger goes on to state :* 

* See also A. Zimmern, The Greek Commonwealth, and B. Farrington, Diodorus 
Siculus (Univ. Wales Press). 



364 Science for the Citizen 

There were no precautions against accidents. The galleries were not propped 
up, and therefore often collapsed, burying workmen beneath them. In ancient 
mines many skeletons have been found of slaves who had lost their lives in 
this way while at work. Nor were attempts made to replenish the supply of 
air or to take other steps for preserving health. When the air in the mines became 
so hot and foul that breathing was rendered impossible the place was abandoned 
and an attack was made at some other point. These conditions must have 
become still more trying wherever, in addition to the mallet and chisel, the only 
other means of detaching the stone was applied, namely fire. The mineral- 
bearing stone was heated and water was then poured over it. There was no 
outlet for the resulting smoke and vapours. This method of constructing 
tunnels and galleries is described by Pliny somewhat as follows : "Tunnels are 
bored into the mountains and carefully explored. These tunnels are called 
'arrugiae,' little ways or little streets. They often collapse and bury many 
workers. When hard minerals occur one seeks to blast them with fire and 
vinegar. As the resulting steam and smoke often fill the tunnels, the workmen 
prefer to split the rock into pieces of 150 lb. or more, and for this purpose they 
use iron wedges and hammers. These pieces are removed from the galleries 
that have been hewn out, so that an open cavern is formed. So many of these 
caverns or hollows are made adjacent to each other in the mountain that finally 
they collapse with a loud noise, and so the mineral in the interior becomes 
exposed. Often the eagerly sought gold vein fails to appear, and the long- 
sustained and arduous work which had often cost many human lives has been 
in vain." . . . The miner of ancient times was nearly always either a slave 
or a criminal. This explains why the means used remained unchanged for thou- 
sands of years. The purpose of machines is to economize labour or time. It 
was not considered necessary to make the work easier for the slave, whose 
hard lot inspired no sympathy, although it kept him to the end of his days 
buried in the gloomy depths of the earth, suffering all sorts of torments and 
privations. There was mostly a superabundance of slaves. After campaigns there 
were usually so many that great numbers of them were massacred. So there was 
no dearth of labour. Time was as yet but little valued. And so it happened that 
in almost all the mines of the ancients only the simplest means were adopted. 
In the copper mines of Rio Tinto and Tharsis in the Spanish province 
of Huelva, which were worked by the Romans and the Carthaginians, the 
method of working was so simple that the slaves in the mines had to scratch 
off with their fingers the clay which covered the ore. The clay which is found 
nowadays in ancient mines still bears the impress of thousands of fingers. 

Quite otherwise were the social foundations of the mining industry when 
civilization spread to northern Europe. Surface work on tin in Cornwall 
and Devon probably continued from the Bronze Age to early medieval times 
on an essentially tribal basis. According to Lewis (The Stannaries) the tin 
workers of twelfth-century England were associations of free men. In time 
these profit-sharing syndicates became transformed into share-owning com- 
panies, employing wage earners. Even so, labour was still scarce, and pene- 
tration to deeper levels in a humid climate was accompanied by constant 
flooding. Coal-mining, which encountered further difficulties on its own 
account, was a new industrial enterprise. Concerning the condition of the 
medieval coal miner, Nef (The Rise of the British Coal Industry) says : 

The modern organization of the coal industry, involving as it does the 
employment of a large industrial proletariat by absentee capitalists, makes it 



The Third State of Matter 365 

easy to assume that the miner has always been a wage earner. When a labour 
leader in our day refers to the miners as the aristocrats of the labour move- 
ment . . . what he almost certainly has not in mind, though it might well 
appeal to his audiences, is the position of the miners in the Middle Ages. In 
medieval England labourers in the metal mines were often granted special 
privileges and immunities of a kind rarely enjoyed by other manual workers. 
. . . Feudal lords in some cases turned over their minerals in return for a 
share of the output, to small autonomous associations of working miners. In 
coal mining "many pits were worked by 'associations charbonieres. 9 " 

Nef hints that coal, like gunpowder, paper, and (probably) the magnet, is 
possibly part of our cultural debt to Chinese civilization. The Chinese appar- 
ently knew its use before the Christian era, and Marco Polo mentions the 
strange practice of the natives who burned for fuel "black stone . . . dug 
out of mountains where it runs in veins." In western civilization its use does 
not seem to have been known before about 1200, at which date a Liege 
chronicler refers to a "black earth very similar to charcoal" used by smiths 
and metal workers. Though export from Newcastle is recorded in the four- 
teenth century, coal was little used before the beginning of the sixteenth 
century, and then mainly from outcrops. By 1597 it was "one principall 
commoditie of this realme." By 1661 we are told that a "hellish and dismall 
cloud of Sea-Coale hangs perpetually over London." Mine-owners could 
no longer afford to throw away life with the recklessness of the Mediterranean 
slave-owning civilizations. How air becomes unfit for human beings to 
breathe, why some air is inflammable and why explosions occur, were 
questions which demanded an answer, and engaged the attention of the best 
brains. Francis Bacon, generally more ready with advice than with remedy, 
proposed a scheme for raising water from "drown'd mineral works." Pumps 
were being introduced to drive fresh air into the tunnels as well as to draw 
water out of them. The wind was no longer blowing whither it listeth. When 
we know how to control anything, we cease to regard it as spiritual. 

This aspect of the social background of scientific discoveries which led to 
the rise of modern chemistry is discussed by Nef (The Rise of the British Coal 
Industry} in the following citation : 

Boyle's interest in experiments of this nature, and the constant discussion 
of the Royal Society concerning the means of increasing the output of various 
commodities, or of heightening the efficiency of different technical processes, 
show a trend of the utmost importance. These "natural philosophers" of the 
English Restoration were economists as well as scientists; and, beside their 
old religion, which they resolved should not stand in the way of their achieve- 
ments, they had begun to erect a new religion the religion of production. It 
was one of Boyle's principal propositions, to which he reverted again and 
again, "that the Goods of Mankind may be much increased by the Naturalist's 
Insight into Trades." . . . There is a clear relation between the appearance 
of the extremely influential British school of "natural philosophers," and the 
growth of the British coal industry. The proceedings of the Royal Society for 
the first thirty or forty years following the grant of its charter in 1660, are full 
of discussions which have a bearing, often direct and perhaps even more 
frequently indirect, upon problems connected with mineralogy, with the mining 
and the use of coal. Colliery owners like Lowther, who was a member, or 



366 Science for the Citizen 

Sir Roger Mostyn, who was not, send in accounts of explosions or of new 
methods of finding coal. Local naturalists send in samples of mineral fuel, or 
of strata resembling it, to be analysed and examined. Sir Robert Southwell 
reads a paper on the advantages of digging canals to supply London with 
cheaper coal. Boyle conducts experiments in order to determine the difference 
between coal and wood. The members meet to consider the projects on foot 
for smelting with coal. They puzzle over the causes for the strange fires which 
occur in the coal seams. . . . Soon after the incorporation of the Royal Society 
the King signified his pleasure that no patent should be granted for any "philo- 
sophical or mechanical invention," until it had been approved by the Society. 
And the problem of invention in the seventeenth century was to a considerable 
degree directly related to coal. "Through necessitie, which is the mother of 
all artes," wrote Howe, in 1631, "they have of very late yeeres devised the 
making of iron, the making of all sorts of glasse, and the burning of bricke, 
with sea coal or pitcoale." In this case the necessity was the substitution of coal 
for wood; in other cases it was the more adequate drainage of the mines, or 
the cheaper carriage of fuel over ground (pp. 253-4). 

THE MECHANICS OF FLUIDS 

Galileo's experiments on the way in which bodies fall overturned Aristotle's 
doctrine without providing any alternative explanation of the familiar experi- 
ences which seem to support it. Our everyday experience of falling bodies 
of leaves blown from a tree, or tiles falling from a roof suggests that the 
rate at which bodies fall depends chiefly on their density or relative "heavi- 
ness." Galileo's experiments showed that this is not true about comparatively 
heavy bodies of compact shape falling through air. To resolve the paradox 
experience of everyday life suggests that we have to reckon with other cir- 
cumstances affecting the motion of falling bodies. Bodies fall under their own 
weight through liquids. We all know that a penny falls faster in water than in 
treacle, just as it falls faster in air than in water. We all know, too, that some 
bodies fall upwards, i.e. float., while others fall downwards, i.e. sink, in a 
liquid, and that bodies like a piece of tin which will sink in water may float 
on a heavier fluid like mercury. In other words, we have to take into account 
two facts which Aristotle rejected. One is that air has weight. The other is that 
like other weighty things it has inertia, i.e. it objects to being pushed to make 
way for falling bodies (p. 375). Galileo's solution of the paradox was that if 
solid bodies are immensely heavy compared with air, the effect of the air on 
the rate at which they fall must be negligible. So a croquet ball as large as a 
cannon ball gathers speed at nearly the same rate. On the other hand, a toy 
balloon as large as a cannon ball is mostly air. Its density is not very much 
greater than air itself. So it falls more slowly. 

Alexandrian science had already supplied the clue, without recognizing its 
worth. One good reason why Alexandrian science failed to take the step which 
would have made chemistry an exact science is suggested by the fact which 
led Galileo to conclude that the weight of air raises water in the same way as 
the weight of mercury (Fig. 230) does. Galileo knew what the miners of his 
time knew to their cost, since it compelled them to raise water by relays of 
pumps and cisterns (Fig. 218). The Greek word for vulgarity was pavavoia, 
which also meant handicraft, especially smithy work. It came from Paw6$> a 



The Third State of Matter 367 




FIG. 218 

(From Professor Wolf's History of Science, Technology and Philosophy in the 
XVIth and XVIIth Centuries") 

This illustration from the Agricola Treatise shows pumps used in relays because 
the weight of the air is only sufficient to raise water about 33 feet. In Galileo's time it 
was respectable to know what every miner knew. The Greek materialists who taught 
that air has weight probably based their conclusion on the same experience of everyday 
life in mining areas. Aristotle, who despised manual work and advocated slavery for 
getting it done, devoted himself to refuting the materialist doctrine. His predilection 
for slavery helped to stop chemistry from further advance for about two thousand 
years, 



368 Science for the Citizen 

forge. Perhaps Aristotle himself would have known that water cannot be 
lifted more than about 30 feet by a pump, if his intellectual powers had not 
been hampered by contempt for work which could be carried out by slaves. 
Those who accepted the Aristotelian teaching relied on the self-evident 
principle that "nature abhors a vacuum" to account for the way a pump 
works. As Galileo sceptically remarked, they could offer no reason why this 
abhorrence stopped short suddenly at a height of 30 feet from the ground. 
In Galileo's Dialogues concerning Two New Sciences the following passage 
explicitly refers to the way in which everyday experience of the world's work 
compelled attention to the weight of the air : 

Thanks to this discussion, I have learned the cause of a certain effect which 
I have long wondered at and despaired of understanding. I once saw a cistern 
which had been provided with a pump under the mistaken impression that the 
water might thus be drawn with less effort or in greater quantity than by 
means of the ordinary bucket. The stock of the pump carried its sucker and 
valve in the upper part so that the water was lifted by attraction and not by a 
push as is the case with pumps in which the sucker is placed lower down. This 
pump worked perfectly so long as the water in the cistern stood above a certain 
level; but below this level the pump failed to work. When I first noticed this 
phenomenon I thought the machine was out of order; but the workman whom 
I called in to repair it told me the defect was not in the pump but in the water 
which had fallen too low to be raised through such a height; and he added 
that it was not possible, either by a pump or by any other machine working on 
the principle of attraction, to lift water a hair's breadth above eighteen cubits; 
whether the pump be large or small, this is the extreme limit of the lift. 

We have seen (p. 243) that Alexandrian science had established what we 
call the fundamental laws of static equilibrium for solids, that is to say, the 
conditions which must be satisfied when two or more weights are perfectly 
balanced. The advanced development of irrigation in ancient times had also 
borne fruit in the discovery of general principles about the equilibrium of 
liquids. The basic principle of equilibrium for liquids depends on a fact of 
everyday observation. If two columns of the same liquid are in direct con- 
nexion and both open to the outside air, they are at rest only when the height 
of each is the same. For practical purposes height in this context usually means 
vertical distance above the ground, but if we consider the water of the oceans, 
it is evident that vertical height implies distance from the centre of the earth. 
The very old device called the siphon (Fig. 219) for emptying cisterns depends 
on the same elementary principle of fluid equilibrium, which is the basis of 
any rational system of water supply for a city. Why the Romans built aque- 
ducts across valleys is still an enigma. It may have been through inability to 
make strong pipes, or it may have been because much of the available know- 
ledge in ancient civilization was wasted through the absence of a system of 
public instruction to diffuse knowledge from one field of human activity to 
another. The principles of irrigation were fully understood by Archimedes 
in 200 B.C., and the feats of hydraulic engineering achieved in dynastic 
Egypt were far superior to those of the Romans. 

If we apply Galileo's way of measuring forces to the force which pulls one 
column up and another down till the water "finds its own level," we can state 



The Third State of Matter 369 

the principle in a way which points to a number of other truths about fluids. 
Fig. 220 shows two columns of a fluid contained in two tubes connected 
by a cross-piece, hence the height (h) of fluid in each is the same. The bore 
of each tube is uniform. The areas of cross-section (A and a) are different. 
The height (h) of fluid above any horizontal level in either column is the 
volume per unit area of cross section (h = v a), and the volume is proper- 





FlG. 219 

The principle that a liquid "finds its own level" implies that a difference of pressure 
between two columns of liquid, free to move vertically upwards, is communicated 
equally in all directions. Thus a tube used to connect two fluid columns of different 
vertical height siphons off the fluid in the higher column till both levels are the same. 
Likewise the pressure of the fingers on a punctured rubber ball filled with water forces 
water out of the pores at all angles to the plane of compression. 

tional to the mass, if the density of both columns is the same. So if the height 
is the same in both columns, the mass per unit area (m -f- a) is the same, and 
the force of gravity acting on unit area (mg -7- a) is also the same in each 
column at the same level. Force per unit area is called pressure. So two 
columns are balanced when the pressure at one and the same horizontal level 
is the same in both. The siphon shows that the connexion between the 



370 



Science for the Citizen 



columns may take any path upwards or downwards. Hence the law of equili- 
brium for fluids also implies that pressure is communicated equally in all 
directions. 

This is the principle of the hydraulic press which has long been used in 



-7JT- 

i 
i 
i 

h 



mass m 
vol ah 



Ah 



m 
a 




f- 
" 



/.Mg= . 

A a 



3uh 



Equilibrium between two connected columns of the same fluid, if both are free to 
move vertically upwards, is attained when both are the same vertical height. This 
implies that the force on unit area is the same at any level in each; hence a column of 
large sectional area will support a proportionately larger weight than a column of 
smaller area. Thus a small force acting on a column of small sectional area will cause 
a connected column of large sectional area to exert a proportionately larger force in 
the opposite direction. 

You will notice also that a large downward displacement of m would be necessary 
to produce a small upward displacement of M. As in the lever, the pulley, and other 
machines, work is the product of the weight and the distance through which it moves . 

the textile industry. If a mass M rests on a float in one limb of area A, 
the pressure of the column below is increased by an amount M ~- A. Equi- 
librium is restored when a mass m is placed on the float in the other limb 
of sectional area a, provided that Mg 4- A = mg -f- a. So a small weight 



The Third State of Matter 371 



applied to a small area will keep up a very large weight resting on a wider 
area in an arrangement like the one shown in Fig, 221. 

Upward thrust of ram 
f c 2Va 




NX 



Force applied 

rrW 



1 






ji 


II 








cL:^ 


^ 


^r^= 


~1 

-ci~ 


^=^1 




<&.(* 








=zr^ 





-Force on plunger 



lw 

Ci 



arm 



FIG. 221. THE HYDRAULIC PRESS 

A second principle of equilibrium for fluids known in ancient times is 
sometimes called the Archimedian principle. If you have ever tried to raise 
yourself while taking a bath you will have noticed that your body seems 
to be lighter when immersed in water than it is ordinarily. The downward 
pulling power on a body immersed in a liquid is therefore less than it would 
be in air. If a weight is immersed in water a smaller weight hanging freely in 
air will balance it. 

The difference is equivalent to the weight of the water displaced. A mass 
M in descending through a fluid is not immersed till it has raised an equivalent 
volume of the liquid (Fig. 222). It has to raise a mass w of liquid equivalent 
to its own volume, as well as the mass m in the scale pan, i.e. a total mass 
(m + w). If the mass in the scale pan balances it, 



or 



M = m + w 

w M m 



A body therefore loses weight in a liquid by an amount (wg) equivalent to 
that of the same volume of liquid. 

According to the Galilean method of measuring it, the pull on the descend- 
ing weight is the product of its acceleration and its mass. It weighs less 
in a liquid because, as we know, it falls more slowly than in air. If we call 
its acceleration when it starts* to fall in water a, its pulling power is Ma. This 
pulling power is balanced by a mass m (= M w) suspended in air. If the 

* Remarks on the parachute in a later paragraph (p. 375) will explain the need for this 
qualification. 



372 



Science for the Citizen 



acceleration of a body falling in air is g, we have a pulling power Ma balanced 
by a pulling power mg, i.e. 

Ma = (M w)g 



Mass M=/rc+w 
Volume V 




Water displaced 

Mass w 
Volume V 



FIG. 222. THE PRINCIPLE OF ARCHIMEDES 

If M is the mass of the body when weighed in air, m its apparent mass when weighed in 
water, and w the mass of the water displaced, i.e. of the equivalent volume of liquid, 

M = m -f w 
or w M m 

If V is the volume of the body (and hence of the water it displaces), its density is 
M ~ V. The density of the fluid is a; -f- V. Hence the ratio of the density of the body 
to the density of fluid is 

M -7- V :w ~V = M ~w 

The ratio of the density of a body to that of water (its "specific gravity") is therefore 
Weight in air 4- (Weight in air weight in water) 

Since w and M are masses corresponding to the same volume, they are in the 
same ratio as the densities (d and D) of the fluid and the descending weight. 

d' 



Galileo's way of measuring forces therefore shows us correctly when a 
body floats or sinks. If the body is very much denser than the fluid, d is very 

d 
small compared with D, and d -f- D is a very small fraction. Hence 1 -=r 

will not diner very much from unity, and the acceleration a of the body 
descending through the fluid will be nearly as great as it would be if it were 
falling through air. If the density of the fluid is nearly as great as that of 

d d 

the body itself, will be a fraction just less than unity, 1 g will therefore 



The Third State of Matter 373 

be small, and a will be small compared with g. That is to say, the body wil 

d . 

fall very slowly. If the body is less dense than the fluid, - is greater than 
i D 

unity, and 1 ^ is a negative quantity. That is to say, the body immersed 

in the fluid will have an acceleration upwards instead of downwards. In other 

/ d\ 
words, it will float. If d is exactly equal to D, ( 1 =r ) is zero, and the 

acceleration is therefore zero. The body neither sinks nor floats. It rises or 
falls with the application of a very small force. The modern submarine is 
equipped with tanks which can be rilled with water or emptied by means of 
a store of compressed air. When the tanks are quite full the total density of 
the submarine is slightly greater than water. The expulsion of a small quantity 
of water is then sufficient to make the submarine rise to the surface. 




Wdgit 




FIG. 223. EXPERIMENTAL PROOF OF THE PRINCIPLE OF ARCHIMEDES 
Two identical weights fit exactly into two cylindrical vessels. When one weight is 
suspended in water from a scale pan, the weight and its vessel in the opposite scale 
pan may be made to balance it by filling the other vessel with water to the brim. 

The same reasoning leads to another conclusion which can be verified, 
namely, the fact that a very small part of an iceberg is visible above the 
water-line. At freezing-point water is 1-083 times as dense as ice. So the 
mass M of ice below the water has an acceleration (1 1 -083)^, and exerts 
a push on the ice above equivalent to (1 1-083)M^ = 0-083 Nig. The 
negative sign indicates that the acceleration and push are directed upwards. 
If downwards, acceleration is denoted by the positive sign. The mass (m) 
of ice above pushes downwards with a force equivalent to mg. And since 



374 



Science for the Citizen 



these forces balance one another, they are equal. Hence w = 0-083 M, 
i.e. M the mass of submerged ice is about twelve times w, the mass of ice 
exposed. 

This gives us a clue for which we are seeking. Aristotle rejected the 
possibility that air has weight because a bladder weighs as much when it 
is blown up as when it is collapsed. We might just as well argue that water 
has no weight because a tin can weighs just as much in water after a steam 
hammer has flattened it out. A bladder inflated with air could only weigh 
more than one which was not inflated if both were weighed in something 
lighter than air., or in an empty space. If air has weight, its weight must be 




FIG. 224 

If we use Galileo's method for measuring force, we can infer the Archimedean principle 
from the observed fact that fluid pressure is communicated equally in all directions. 
A cylindrical mass (M) of density D., height H, uniform sectional area A, and volume 
V(== AH), is held in position by another mass (m) suspended in air at the same distance 
from the fulcrum from the other arm of a balance. The surface of a fluid of density 
d is just level with the top of the immersed cylinder of mass M. The force acting 
downwards on the mass m is mg. The force acting downward on the mass M would be 
Mg(= VD^), if it were not opposed by the upward push of the fluid acting on the 
base with a force per unit area Hdg. Hence the total upward force on the base is 
AHdg Vdg, and the weight mg is therefore balanced by a net downward pull 
VD^r Vdg> i.e. 

mg = VDg Vdg 

... \7T\ \7 sJ 

m = \JLJ- \ a = 

Since density is mass per unit volume (i.e. M- 

m = M( I ~ 



extremely small compared with that of most solid bodies and liquids, and we 
can accommodate Galileo's experiments with the slow initial descent of a toy 
balloon without difficulty by a very small modification in the original form 
of Galileo's principle. Suppose that bodies fall through empty space near the 



The Third State of Matter 375 

earth's surface with constant acceleration g. If a is the acceleration with which 
a solid body falls in air, it will rarely differ appreciably from g, because the 
density D of most solid bodies is very large compared with d the density of 
air. This is obviously not true of a toy balloon. It is largely composed of air 
when it is blown up. Its density D considered as a whole is not very much 




= M(i- g )g 



_ 

M "" D 
FIG. 225. THE ICEBERG 

The balances indicate the forces necessary to balance the downward pull and upward 
push of the two parts. For convenience of drawing the attachments are at the sides. 
Of course tilting will occur unless the attachment is vertically in line with the centre of 
gravity. 



greater than d, and f 1 .=- j is a small fraction. Hence it falls slowly. A gas 

less dense than the surrounding air (e.g. hotter air or coal gas) must have a 
negative acceleration for analogous reasons, and will therefore rise upwards. 

The behaviour of a parachute illustrates the fact that air has another charac- 
teristic of weighty matter. It is sluggish, and this sluggishness or inertia gives 
us a clue to the propagation of sound (p. 321) and to certain peculiarities of 
falling bodies. On account of its inertia air resists any attempt to push it out of 



376 Science for the Citizen 

the way. This resistance to displacement naturally depends to some extent on 
the shape of an object. It is relatively great if a moving object presents a large 
surface relative to its bulk, and this fact is the basis of streamline designs for 
fast cars. What is not easily recognized from our everyday experience of motion 
is that air resistance and that of any fluid medium is greater at greater speeds. 
Roughly speaking, air resistance is proportional to the square of the speed of a 
body moving in it. So when a body has been falling through air for some time, 
there comes a time when the resistance which it encounters just balances its 
own weight. Thereafter it gains no more speed. It continues to move at a fixed 
speed. On account of its shape a parachute has a high initial resistance. Since 
it traps a lot of air, its effective density is small. Hence its initial acceleration is 
also small. It reaches the limit of its power to gain speed very quickly. 

If he did not cut the cord to release the parachute after jumping from his 
plane, an aviator would continue to fall through a relatively long distance while 
gaining speed at a rate differing little from the fixed acceleration of bodies 
falling in a vacuum. At a certain limit his acceleration would fall off rapidly, 
reaching a "terminal" fixed speed many times greater than that of a para- 
chutist. By the time he was travelling 170 miles per hour, he would have 
ceased to gain speed. In the Galilean sense, he would have no effective weight. 
Why then would he be mangled? An everyday experience provides the 
answer. If an egg is dropped from a height of half an inch, it is not broken. 
If it is dropped from a height of six feet it is smashed. Within these limits its 
Galilean weight does not differ sensibly. So its Galilean weight is not directly 
responsible for disaster. The egg shell has internal inertia, and extensibility. 
It cannot communicate motion throughout all its parts instantaneously. When 
the bottom touches the ground, the top is still moving. So some parts are losing 
speed more quickly than others. At the moment of impact the distribution 
of accelerations is not uniform, and these accelerations correspond to Galilean 
forces. Thus there is an unequal distribution of internal stresses which in one 
direction or another exceed the breaking point. How big these internal stresses 
are depends on whether the speed immediately before impact is great or small 
compared with the rate at which the material of the shell can communicate 
motion. The surface of a mouse is much greater relative to its weight than that 
of a horse. Hence it encounters much greater air resistance and has a much 
smaller terminal or limiting speed of fall. For this reason it can fall through a 
much greater height without being killed. 

When von Guericke and Boyle had devised a pump for sucking the air out 
of a vessel, Newton was able to show that a guinea and a feather keep pace in 
falling through a vacuum. Proof that air has weight first came from another 
source. Before explaining how the discovery was made, two empirical appli- 
cations of the Archimedean principle in everyday life may be noted. The 
density of some saleable fluids (e.g. milk and spirits) is controlled by legisla- 
tion, and the density of the fluid in storage batteries is a useful indicator for 
testing when they are fully charged. For routine determinations of this kind, 
accurate weighing of an accurately ascertained volume of fluid would be much 
too laborious, and is replaced by the device known as the hydrometer 
(Fig. 226). This is essentially a metal tube or a glass tube with some mer- 
cury at the bottom to make it sink till the weight of fluid displaced is equiva- 
lent to the weight of the instrument. The volume of fluid whose weight is 
equivalent to weight of the instrument depends on its density. So the hydro- 
meter sinks more if the density is lower. A graduated scale records the length 



The Third State of Matter 377 



of the submerged part. The divisions marked off correspond to the levels 
which fluids of particular densities reach. Only relative measurement of 
densities is made in this way, water being taken as the standard. Density re- 
ferred to water as the standard (d = 1) is called specific gravity.* 




FIG. 226 
The scale marks show how deeply it sinks in fluids of different density. 

A second application of the principle of buoyancy is the "PlimsoU" line. 
An iron ship floats because the weight of the shell is far less than the weight 
of the total volume of water displaced. Every addition of cargo makes it sink 



FW 




WNA 



FIG. 227 

The Plimsoll line on all ships in Lloyd's register, L.R. The several marks are the 
loading limits for fresh water, Indian summer seas, temperate seas in summer and 
winter, and winter in the north Atlantic. 

further till an equivalent weight of water is displaced. It is not safe to load 
a ship so heavily that the addition of a small amount of extra ballast will 
sink it. The limit of safety depends on the density of the water, which varies 

* If volume is measured in cc. and weight in grams, the density and the specific 
gravity of the same substance have the same numerical value at 4 C., since 1 c.c. of 
water at this temperature weighs 1 gm. 



378 



Science for the Citizen 



appreciably according to its salt content and temperature. Sea water is denser 
than fresh water, as we know if we are swimmers, and the warm tropical 
waters are less dense than Arctic seas. The law now enacts that all vessels 
carry a scale like the scale of a hydrometer, showing the different levels 
beyond which the water level must not be allowed to pass when loading in 
fresh or sea water, arctic or equatorial seas (Fig. 227). 

THE PRESSURE OF THE ATMOSPHERE 

The most direct way of proving that air has weight is to compare the weight 
of an exhausted vessel with its weight when filled with air. When the temper- 
ature is 60 F. at sea-level, the weight of one litre (1,000 c.c.) of air is approxi- 
mately 1 J grams. That is to say, a vessel of 1 litre capacity (roughly one-fifth 





FIG. 228 

Two simple experiments illustrating the weight of the atmosphere, In the right-hand 
one, the pressure of the air, communicated like the pressure of the water in the siphon 
in all directions, keeps a full tumbler sealed by a piece of paper placed across the 
open end. 

of a gallon) weighs 1| grams (about a twentieth of an ounce) less when 
exhausted. The discovery that air has weight did not result from, but pre- 
ceded, the discovery of the vacuum pump. It was made by Torricelli, a pupil 
of Galileo, about the year A.D. 1643, and was really a miniature demonstration 
of the principle which underlies the necessity of raising water in relays 
(Fig. 218). According to Burnet, the Ionian Empedocles correctly interpreted 
the raising power of the common pump as the weight of the air pressing on 
the source of water. If so, the early Greek materialists had solid ground for 
their faith in the indestructibility of matter. Be that as it may, Aristotle's 
teaching triumphed, and, so far as we know, was accepted by the Alexan- 
drians, whose knowledge of the mechanics of fluid should have sufficed to 
exhibit the flaw in his reasoning. 

An old trick which is easy to repeat with a tumbler of water and a wash 
basin may have suggested the experiments of Torricelli. If we turn a tumbler 
upside down under water and draw it upwards, the water does not descend 
inside it, so long as the rim is just below the surface of the water. If we 
place a piece of paper on a tumbler brimming over with water, we can turn 
it upside down in air without emptying it. The paper remains apparently 
glued to the edge. In performing similar experiments, in which a cy Under 



The Third State of Matter 379 

filled with mercury was inverted over a trough containing mercury,, Torricelli 
found that the fluid descends a certain distance if the cylinder is sufficiently 
long, leaving a space above it without any sign that bubbles of air have 
passed up. The vertical height of the mercury column with the empty space 
(Torricellian vacuum) above it was not affected by tilting the cylinder side- 
ways, or by using tubes of different sectional area. At sea-level the height 
of the mercury column above the level of mercury in the trough is 760 mm., 
or roughly 30 inches. A variation of the original form of the experiment is 
to fill with mercury a U-tube of which one limb is closed, holding it in a 
nearly horizontal position, taking care to let in no air bubbles. If the other 
limb is sufficiently long the mercury sinks in the closed one when the U-tube 
is held upright till the vertical difference between the two columns is 760 mm. 
(at sea-level). Either arrangement is what we now call a mercury barometer 
(Fig. 229). 

We have seen that pressure at any horizontal level in a fluid is the force 
exerted by the weight of the overlying fluid on unit area (mg ~ a). Since mass 
is the product of volume (v) and density (d), 

pressure = vdg ~ a, 

and since volume per unit area (v -~ a) is the height (K) of the overlying 
column, the pressure or force per unit area of a column of fluid on its base 
is hdg. To put it in another way, a pressure hdg is required to raise a column 
of fluid to a height h. For measuring what supplies this pressure, the equili- 
brium of two liquids of different density (Fig. 229) furnishes a precise parallel. 
Two connected columns of fluid are only in equilibrium when the vertical 
height of both is the same, providing that their densities are the same. If a 
straight open tube dips into a trough of mercury, the mercury in it rises 
above the level of mercury in the trough when water is poured on to the 
latter. Similarly, if water is poured into one limb of a U-tube containing 
mercury, the mercury in the other rises a little. In either case the more general 
form of the law of equilibrium is true. At any level in the mercury the pressure 
of both columns above it is the same, e.g. at the level where the mercury 
(density D) is in contact with water (density d\ the heights of the overlying 
columns of water (h) and mercury (H) are such that 

HDg = hdg 
or 

/. H~h = d-^D 

This means that if mercury is 13 \ times as dense as water, a column of 
mercury could just be supported in a vertical position by the pressure of a 
column of water 13 J- times as long, in the same way as the water in the 
tumbler is supported in a vertical position by something pressing on the 
paper and communicating its pressure upwards and downwards in all 
directions equally. 

In these experiments we have water on one side of a mercury column, 
and no water on the other. The pressure exerted by the weight of the water 
maintains the difference in level between the two columns of mercury. In 
the barometer we have air on one side and apparently none on the other. 
Is the pressure supplied by the fact that the air has weight? A simple experi- 



380 



Science for the Citizen 



ment completes the analogy. If we withdraw some of the water in the last 
experiment (Fig. 229), the height of the water column is less, and the differ- 
ence between the two levels of mercury is diminished. We can make the 
height of the air column pressing on a mercury barometer less by taking 
the barometer to the top of a mountain. This test was devised by Pascal 



-Noai^r 
(yaruuni) 



IF 3 



-Air 




Water 



FIG. 229 

Above, two forms of the barometer. Below, the weight of a column of water, pressing 
on mercury (density D) with a force HDg = hdg per unit area, lifts a column of mercury 
to a height H in the same way as the weight of the atmosphere maintains the mercury 
level of the barometer. 

(about 1648) soon after Torricelli's experiments, and is now used as a means 
of calculating altitude. At the top of Mount Everest (5| miles up) the weight 
of the atmosphere will only support a column of mercury 11 inches high. 
At the height of 9-6 miles, reached by Professor Picard in 1931 (Fig. 241), the 
mercury barometer only registers 5 inches. That the same force acts on any 
liquid is easily proved. Since mercury is 13| times as dense as water, the 



The Third State of Matter 381 

pressure which supports a column of mercury 30 inches long at sea-level 
would support a column of water 13| times as high, i.e. about 34 feet. 
This is easily shown to be true, and provides a good enough reason for pre- 
ferring the use of mercury to water in constructing a barometer. There are 
several others. 

The most familiar use of the barometer in everyday life depends on the 
fact that water vapour is less dense than air. So a mixture of water vapour 
and dry air is less dense than pure air. Since the height of the atmosphere 
may be taken to be nearly constant, changes in the atmospheric pressure at 
any particular place chiefly depend on moisture and temperature. A fall of 
pressure means that the air is less dense, and this generally means that it 
is more moist. Hence rain is to be expected. This rule is not highly reliable, 
because changes in pressure also result from unequal heating of the earth's 
surface. Warm air is less dense than cold air, and the atmospheric pressure, 
therefore, depends partly on the temperature. It is constantly changing 
because of the winds so produced. A further complication arises from the 
fact that more water vapour is required to saturate warm than to saturate 
cold air. In an island climate, the direction of the wind is specially important. 
In England rain may be anticipated when cold winds are blowing over the 
sea from the north-east, even if the barometer is rising. In the weather 
forecasts now published, most importance is attached to the relative distri- 
bution of pressure based on wireless signals which make it possible to record 
how zones of high and low pressure are shifting from day to day. A chart 
constructed on the basis of simultaneous records of pressure at different 
stations will frequently show closed areas of high pressure (anti-cyclones) 
from which dry winds are blowing spirally outwards, or closed areas of low 
pressure (cyclones) where wet winds are blowing spirally inwards. Their 
position changes from hour to hour, so that it is possible to foretell within 
short limits where they will next be. The approach of a cyclone betokens 
wet weather, that of an anti-cyclone dry weather. 

At sea-level in our climate the height of the mercury column varies between 
29 and 31 inches. Since the force acting on unit area at the base of a column 
of fluid (hdg) only depends on its height, density, and the value of g which 
varies only very slightly with latitude and altitude, it can be calculated at 
once from the height of a column of fluid of known density. The mass of 
1 cub. ft. of mercury is 848 Ib. The weight of a column 1 sq. ft. in cross- 
sectional area and 30 inches high is 848 x 30 -~ 12 = 2,120 Ib. So the 
pressure of air which supports it is equivalent to a weight of 2,120 Ib. on 
every square foot, or approximately 14f Ib. per square inch. A barometer 
can also be adapted as a pressure gauge (Fig. 230) in measuring low or high 
air pressure when exhausting a space, compressing a gas, or recording the 
pressure of a water supply. Pressure is often measured in "atmospheres," 
the standard or unit (one atmosphere) being the mean pressure at freezing 
temperature in latitude 45. This is 760 mm., or 29-92 inches of mercury, 
or 33 feet of water. Thus the pressure of a town water supply that will raise 
water in an upright tube 100 feet high is almost exactly 3 atmospheres. 
That is to say, the force exerted on every square inch is equivalent to a weight 
of 3 x 14 or 44 Ib. to the nearest pound. 



382 



Science for the Citizen 



THE SPRING OF THE AIR 

To be useful, a recipe for making substances must include the quantities 
of the ingredients and the yield. The importance of chemistry in modern life 
depends on the fact that it can give us rules for finding recipes of this sort. 
Modern chemistry rests on the doctrine of the Greek materialists. In chemical 
processes no matter is lost. Increase in the weight of one ingredient of a 



vaciuim 





FIG. 230 

Two types of pressure gauge. The upper one is for measuring small pressures in the 
neighbourhood of a vacuum. The fluid does not begin to fall until the pressure in the 
chamber to which the open end is attached falls below that of a column of fluid 
of length H; i.e. the gauge only measures pressures less than H. When the fluid 
begins to drop the difference h measures the actual pressure in the chamber. The 
lower one is for measuring small pressures in excess of that of the atmosphere 
(e.g. that of the gas supply). The difference h is the excess of pressure over that of the 
atmosphere; i.e. h must be added to the height of the barometer reading to get the 
pressure in the chamber. Either type of gauge may be made more sensitive by using 
a light fluid like water instead of mercury. Of course, pressure measured in lengths 
of a column of fluid means nothing unless the fluid is stated. If water is used for the 
lower type the barometer reading which must be added to h must be given in water 
units, i.e. 13-6 times the reading of the mercury barometer. 

mixture is accompanied by loss of weight of others which go to its making. 
The science of antiquity knew no general rules for obtaining a good yield. 
In the ancient chemical arts of metallurgy, medicine, and fermentation, 
substances appeared to gain weight and lose it according to no detectable 
rules. The Arab chemists knew that the oxide ore or calx produced when a 
metal is heated in air is heavier than the metal from which it is formed. When 



The Third State of Matter 383 



the oife was reduced in the absence of air by heating with charcoal, the latter 
disappeared, leaving only metal lighter than the original calx. Why this 
should happen they could not see, because they did not realize that air is 
ponderable matter. From the beginnings of the Iron Age, no substantial 
progress in the understanding of chemical processes was made, or was pos- 
sible till air had ceased to be a spirit. When first carried out the experiments 



Wwdpipa 




ar 



'diaphragm' 



< Cord far cLzpress- 

g " 



so 




FIG. 231 MEDICAL ADVANCES IN THE SOCIAL CONTEXT OF THE COMMON PUMP 

When the mechanism of the common pump was fully undcrstood/the old belief that 
respiration allows the vital spirits to escape from the body made way for the recogni- 
tion that the chest movements are simply a mechanical device to ensure that the 
spongy cavities of the lungs whose thin walls are abundantly supplied with blood 
vessels receive a constant supply of fresh air. The fundamental peculiarity of the third 
state of matter is that it spreads out in all directions to occupy all the empty space 
available. Among his many experiments on the mechanics of the pump, von Guericke 
showed that a balloon can be expanded till it burst, if enclosed in a large air space 
from which the air is pumped out. The experiment can be varied by using the lungs 
of a freshly killed sheep. If the windpipe is made airtight by a ligature the lungs can 
be inflated to bursting point. In other words, the space which a gas occupies depends 
on the external pressure to which it is subjected. An alternative modification of von 
Guericke's experiment shown in this figure illustrates what happens in normal 
breathing. The windpipe is open to the air. The cavity of the glass bell jar in which 
the lungs are suspended is airtight, and the floor of the bell jar is closed by a rubber 
sheet which can be depressed by a cord, thereby increasing the capacity of the over- 
lying space containing the lungs. When this is done the lungs can be rhythmically 
inflated and deflated by the up and down movements of the cord. This corresponds 
to the action of the muscular diaphragm or midriff which separates the chest cavity 
or thorax from the abdomen. During inspiration the midriff is depressed so that 
fresh air rushes through the windpipe into the lungs. When it is elevated during 
expiration, some of the foul air is expelled. The internal capacity of the chest cavity 
is also increased during inspiration by the expansion resulting from the contraction 
of muscles which lie between the ribs and force the latter to bulge outwards. 



Science for the Citizen 

of Torricelli and Pascal, now almost commonplace in thek simplicity, were 
the focus of one of the greatest revolutions in human knowledge. They made 
ak a weighty matter. 

By themselves, all these experiments showed was that air has weight. 
They tell us what the weight of the whole atmosphere is, but not the weight 
of a given quantity of ak. This was made possible by a discovery which 
followed immediately afterwards. In the social ckcumstances of Galileo's 
generation one of the pressing technological problems which attracted atten- 
tion was how to pump water out of mines, and fresh air into them. The 
mechanism of the pump depends on what Hooke called the "spring of the 
ak." The discovery of the vacuum was a necessary result of a correct under- 
standing of how pumps work. Although most of us think that we know all 
about how a pump works, there is much to be learnt from it. So far we have 
only considered the force a fluid liquid or gas can exert by its own weight 



-^ 

H 



____ ___ 



,P = 



X 



FIG. 232 

The pressure at the base of a column of fluid of height h is measured by hdg only if the 
column is free to move upwards. 

downwards. The measurement of pressure by the product hdg implies that 
the fluid is only prevented from moving upward by its own weight. In a 
vessel closed at the top this is not so. Since a fluid communicates pressure 
in all directions the pressure of fluid at the base of the projecting arm in 
Fig. 232 is the same as the pressure at the same level in the main body of the 
fluid, and this does not depend on the height of the fluid in the closed limb. 
Similarly, solid bodies do not merely exert pressure in virtue of thek weight. 
They also exert pressure against any attempt to change thek shape or volume, 
and hence density. If we stretch a piece of spring it returns to its normal 
position when we let go. If we press on the handle of a bicycle pump when 
the orifice is closed, it springs back when we release our grip. The pressure 
of the air within it is greater than that of the outside air when its volume 
is diminished. Conversely, it is less than that of the atmosphere when its 
volume is increased. The action of the once "common pump" depends on 
the same fact (Fig. 233). Raising the piston increases the volume of ak in the 
cylinder, and the water rises because the pressure of the atmosphere in the 
well is greater than the pressure of the ak in the pump itself. 
A difficulty which prevented people from looking on ak as a material sub- 



The Third State of Matter 385 

stance may have been partly due to the characteristic which specially distin- 
guishes the third state of matter from solids or liquids. In the gaseous state 
matter is highly compressible. Liquids and solids only suffer minute changes 
of volume under tremendous pressure. In ordinary conditions each has its 
own characteristic density which is fixed. So, when we speak of a quantity 
of something in everyday life we do not need to draw a sharp distinction 
between mass and volume. We measure groceries in pounds or pints 
according to convenience. A pint of bran or meal or beer represents as 
definite a quantity of matter as a pound to us. Nothing we do to a pint of 
beer by ordinary methods at our disposal makes it less than a pint, and two 



t 




FIG. 233. THE COMMON PUMP 

Any pump which continuously forces or sucks up a fluid must have two valves. In 
the figure one valve is the leather washer of the piston. The lips of this are kept closed 
by the pressure of the air in the upstroke. The other is a metal trapdoor kept tightly 
shut by the pressure of the water in the downstroke. 

pints weigh twice as much as one. Unless we know the state of compression 
of a gas nothing of the sort is true. Two pints may weigh less than one pint. 
When we stretch a spring its mass remains constant, but its mass per unit 
length diminishes. So to deduce its mass from its length we need to be told 
what is the stretching force, and how its length varies with it. To deduce 
anything about the mass of a gas from its volume we have to know how the 
density of a unit of mass, i.e. its volume, depends on the pressure. There are 
two very good reasons why it is important to know this connexion. We usually 
measure liquids and powdery solids by volume, because it is generally 
easier to measure volume than to measure mass. This is specially true of 
gases, because they are so light in ordinary circumstances. So no advance 
towards an understanding of how the weights of gases and solids are con- 
nected, when they enter into combination, could be rapid, until it was possible 
to calculate the weight of a gas from the volume it occupies at a particular 
pressure. To get an idea of how much the progress of chemistry depends on 
knowing the spring of the air, you have only to imagine how many people 
would get served before closing-up time if all beer were weighed out on 

N 



386 



Science for the Citizen 



scales before being sold. Another fact which made it important to determine 
the spring of the air was that we cannot calculate how the barometer is 
affected by altitude unless we know it. The atmosphere gets more rare as 
we ascend. If the density of the air were the same at all levels we could 
calculate pressure at any altitude from the product hdg. Since the lower 
layers of air are pressed under the weight of all the air lying above them, 
they are more dense. 

Discoveries, like misfortunes, rarely (if ever) come singly. They have 
their roots in the social experience of the time. The law of the spring of 
the air was independently discovered by Boyle and Hooke in England, 




FIG. 234. BOYLE'S EXPERIMENT 

The outside pressure is H 3 the difference of pressure between the air in the closed and 
open columns is /. So the pressure in the closed column is / + H. 

and by Mariotte in France, within a few years of Pascal's experiment. In 
England it is called Boyle's law, though the credit is probably due to Hooke. 
To discover it, he used a U-tube containing mercury, one limb being 
dosed (Fig- 234). If the bore of the tube is uniform, the volume (v) of ak 
in the closed limb is directly proportional to the length of the air column L. 
When the level of mercury in both limbs is the same, the ak in the closed 
one (A) is not compressed. Its pressure is therefore equal to that of the 
atmosphere (H), (i.e. that on the base of a column of mercury roughly 
30 inches high). If the limb A were in communication with the ak the 
mercury would rise to the same level in both limbs when more was added 
to limb B. If it is closed, the mercury does not rise so far in limb A, and 
the difference of pressure is proportional to the difference (/) between the 
heights of the two columns. So the total pressure (p) of the gas in the closed 



The Third State of Matter 387 

limb is measured by the (7 + H). Since pressure increases when volume 
diminishes, the simplest rule to explore is that the pressure varies inversely 
as the volume, i.e. 

k 



or 

pv = k 

Since L is proportional to v and (7 + H) to />, this is true if 

L(7 + H) = constant 

Some data from one of Boyle's experiments (1662), when the barometer 
reading was 29 inches, show how closely this rule was found to be true. 



L 


' 


L(7 + 29 rJ 


12 





349-5 


11 


2yf 


351-3 


10 


6ft- 


353-1 


9 


10 A 


353-3 


8 


15ft- 


355-5 


7 


2 1 fy 


352-2 


6 


29JJL 


352-9 


5 


41-ft- 


353-4 


4 


58ft 


349-0 




1 b 




3 




352-7 



Taking the mean value of the product as 352-3, you will find it instructive 
to test the limits of accuracy by calculating / from observed values of L, thus 



/. 1= (352-3 -f-L)-29A 

The largest discrepancy, when L = 4, gives 7 = 88-075 29-125 = 58-95 
as compared with 58 125, an error of less than 1 J per cent. Over a wide 
range the experimental error is very much smaller when the best modern 
glass and a more finely graduated scale is used. For most practical purposes 
the connexion between the pressure and volume of gas is therefore given 
by the formula 

pv = k 

Since volume is inversely proportional to density, this is equivalent to saying 
that pressure is directly proportional to the density of air (or other gas), i.e. 



or that density is directly proportional to pressure, i.e. 



388 



Science for the Citizen 



Assuming that the temperature at which all experiments are carried out 
is the same, Boyle's law tells us how to calculate the density of a gas at any 
given pressure. Hence we can calculate the mass of a given volume at a 
particular pressure, once its density has been found at some standard pressure. 
The densities of gases are usually given at the standard pressure of 760 mm. 
of mercury. If a gas occupies a vessel whose capacity is 38 c.c. at 750 mm., 
and at the temperature of freezing water (0 C.), 750 x 38 = k. If its 
volume at C. and standard pressure is V, 760 V = k. Hence 






up&trokz 



dcmmsiroke 



FIG. 235 

The bicycle pump and tyre, like the common pump for raising water, have two 
valves. The leather washer valve of the piston is kept closed by the greater pressure 
in the cylinder during the downstroke, and the rubber tube valve of the tyre is kept 
closed by the greater pressure of the air in the tyre during the upstroke. 

So if its density at standard pressure and C. is 1-3 grams per litre, (1,000 
c.c.), the mass of 38 c.c. at 750 mm. and C. is 

(1-3 X 37-5) -f- 1,000 = 0-04875 grams. 

As we all know, hot air rises. The density of air or any gas diminishes 
considerably as the temperature is changed, if the pressure is kept the same. 
So Boyle's law by itself is a safe guide for measuring gases only if all experi- 
ments are done at the same temperature. Mariotte used it to calculate (see 
Fig. 241) the pressure of the atmosphere at any height above sea-level, by 
allowing for the changes in density due to the compression of the lower layers 
of air by the ones lying above. 



The Third State of Matter 389 



THE VACUUM 

To measure gases by means of Boyle's law in the way illustrated by the 
example, we need to know their densities at some standard pressure and 
temperature. This can be done once for all by weighing a vessel of known 
capacity when filled with the gas, if its weight has been found when it contains 
no gas at all, like the empty space above the mercury in a barometer. A great 
advance was made when the common pump was adapted for creating a 
vacuum. The invention, for which von Guericke seems to have priority 
over Boyle, was made about the same time as the discovery of the spring of the 
air, and like it happened independently in England and on the Continent 






cioumstroke 



upstroke 



FIG. 236 

A vacuum pump may be made from a bicycle pump by reversing the position of the 
valves. 

about the same time. If Boyle is justly called the father of modern chemistry, 
von Guericke might with equal justice be called the father of steam power. 
A device with which he used the suction of air in filling a vacuum to raise 
weights was without doubt the parent of the mechanisms which led to the 
construction of the Newcomen engine. 

The invention of the vacuum pump also made it possible to submit 
Galileo's doctrine of terrestrial gravitation in the modified form, which has 
been stated earlier in this chapter, to the decisive test which Newton applied. 
By showing that a feather and a guinea fall through a long glass cylinder at 
the same rate when all the air is drawn out of it, Newton furnished direct 
proof that bodies fall with constant acceleration in vacua . The constancy with 
which compact heavy solids fall, as in Galileo's experiment, through air is 
not remarkable when we remember how bodies gain speed in falling through 
a fluid. The vacuum pump made it possible to show then that the weight of 



390 



Science for the Citizen 



air at room temperature is only about 75 Ib. per 1,000 cub. ft. An equal 
quantity of glycerine weighs about 1,000 times, and of water about 800 times, 
as much as air at atmospheric pressure. Neglecting air resistance, this means 
(see p. 372) that a substance with the density of glycerine would fall through 
air at an acceleration (1 - T^FO) times its acceleration in a vacuum. So the 
value of g calculated from observations in air would only differ by one part 
in a thousand from g measured in vacua. A solid piece of indiarubber has 
about the same density as glycerine, and if it happens to be a golf ball, falls 




FIG. 237. DEPRESSION OF THE BAROMETER COLUMN BY 
AN AIR BUBBLE 

In making a barometer or pressure gauge, it is important to exclude small air bubbles 
between the mercury and the glass. The presence of bubbles exerts a pressure in the 
dead space, depressing the mercury column below the level to which it would rise if 
the vacuum were perfect. The height of the column measures the difference in pressure 
between the atmosphere (pressure P) and that of the dead space. If the latter is a 
vacuum the pressure in the dead space is 0, and the difference P ~ P. If the 
dead space is not a perfect vacuum : 

difference in pressure (p) = pressure of atmosphere (P) pressure in dead space. 
.'. pressure of air in dead space = P p 

Suppose e.g. that an air bubble of x c.c. at pressure P is introduced into the dead 
space of a tube of uniform sectional area a sq. cm. By the law for the spring of the air, 
Px k. If the dead space is increased from D cm. to d cm. and the pressure drops 
from P to p cm., d D = P p. 



The volume occupied by the air bubble at the new pressure (P p) is ad 

:. (P - p)ad = k - Px 

.: <*(P p) (P p 4- D) = Px 

The only unknown quantity in this equation is p, if the dimensions of the tube, size 
of the air bubble, and the external pressure, have been measured. Hence the new 
height of the column can be calculated. 



The Third State of Matter 391 

like the balls in Galileo's experiments. If it is made in the form of a football 
bladder, which can be blown up till the volume of air it displaces is several 

hundred times as great as the rubber itself, =? may be very nearly equal to 



unity, and ( 1 1 very nearly equal to zero. So it will fall very slowly. 




FIG. 238. RELATION OF THE AIR SPACE IN A DIVING BELL TO THE 
DEPTH OF IMMERSION 

If the sectional area of the bell is A, the dead space is Ab, and the total capacity is Ac. 
Hence the relative volume of the air space is b ~ c. The pressure of water at the 
depth d is equivalent to atmospheric pressure P (cm. of mercury), together with the 
weight per unit area of d cm. of water. Since mercury is 13-6 times as dense as water 
d cm. of water is equivalent to d -f- 13-6 cm. of mercury, and since water transmits 
pressure equally in all directions the pressure in the dead space in equilibrium with 
water at the same depth in cm. of mercury is 

d 



P + 



13-6 



cm. 



Hence according to Boyle's Law, 



I3-6P 



" I ~ 13-6P + d 
It the depth is very great we can put a instead of d. 



THE REVIVAL OF ATOMISM 

Whatever credit may be due to Aristotle as a student of animal life, his 
treatise on physics was neither original nor in advance of his time. His 
Attic partiality for disputation triumphed over the robust common sense 
of his Ionian predecessors in an attempt to accommodate Plato's idealism 



392 



Science for the Citizen 



with the claims of natural enquiry to rank as a gentlemanly pastime. In 
the end the supremacy of Plato's mysticism was the death-blow to Greek 
natural science. Cyril Bailey's collection of the remnants of Ionian know- 
ledge show that the earlier Greeks had an uncanny gift for drawing correct 
conclusions from careful observation of nature without elaborate equip- 




t Rarefied 
air 



H-fc 



Deiistty'D \ ^ 




Density d 



FIG. 239. MEASUREMENT OF DENSITY BY FLUID LEVEL 

On the left are two tubes immersed in the same fluid of density D. The dead space of 
one, in which the mercury level is H cm., is a vacuum; and HDg is the difference of 
pressure between the atmosphere and a vacuum, i.e. the atmospheric pressure in units 
of force. The dead space of the other, in which the mercury level is h, contains some 
air. The difference in pressure between the dead space and the outside air is hDg. 
Hence if P is the pressure inside the dead space, HDg P = hDg and P HDg 
hDg = Dg(H h). Since D is the same if the same fluid is used and g is the same 
at the same locality we can express the pressure difference between two lots of gas by 
the difference of height (H h) of two pressure gauge readings. If some air is sucked 
out of a U-tube dipping into two different liquids, one of density D rising to level L, 
the other of density d rising to level /, the difference between the pressure of air P 
in the closed space and that of the outside air (A) is A LDg as measured by the 
rise of fluid in the left-hand limb, or A Idg measured by the rise of fluid in the 
right-hand limb. Hence 

A - -LDg = A - Idg 
"LDg - Idg 

L _ d 

T~D 
i.e. the densities are inverselyjproportional to the heights of the columns. 

ment such as we possess. The doctrine of Atomism which Aristotle applied 
all his ingenuity to overthrow was more than a lucky guess. It was the fruit 
of close observation by men in close contact with the world's work. 

Although dominated by Plato's teaching^ which treated the shadow world 
of everyday experience with contumely, the revival of classical learning had 
one salutary effect on scientific enquiry. It led to the study of Plato's antagon- 



The Third State of Matter 393 

1st, Epicurus, who had adopted the Atomic doctrine of Leucippus and 
Democritus. Prominent among the translators and commutators of Epicurus 
was Gassendi, who was one of the founders of the Paris Academy. Gassendi 
was the irreverent ecclesiastic who first determined the speed of sound in air. 
He also carried out an experimental demonstration which led to the abolition 
of witch-burning in Catholic France more than half a century in advance of 
Protestant Scotland. The incident is recounted in Robertson's A Short 
History of Freethought Ancient and Modern (Vol. II), as follows : 

Among his other practical services to rationalism was a curious experiment, 
made in a village of the Lower Alps, by way of investigating the doctrine of 
witchcraft. A drug prepared by one sorcerer was administered to others of the 
craft in presence of witnesses. It threw them into a deep sleep, on awakening 
from which they declared that they had been at a witches' Sabbath. As they 
had never left their beds, the experiment went far to discredit the superstition. 
One significant result of the experiment was seen in the course later taken by 
Colbert in overriding a decision of the Parlement of Rouen as to witchcraft 
(1670). That Parlement proposed to burn fourteen sorcerers. Colbert, who 
had doubtless read Montaigne as well as Gassendi, gave Montaigne's prescrip- 
tion that the culprits should be dosed with hellebore a medicine for brain 
disturbance. In 1672, finally, the king issued a declaration forbidding the 
tribunals to admit charges of mere sorcery; and any future condemnation were 
on the score of blasphemy and poisoning. 

Epicurus was not himself a scientific investigator. He was interested in 
man's social life. Like Karl Marx (who wrote his doctorate dissertation on 
the Greek atomists), he regarded the teachings of natural philosophy as the 
proper basis for social conduct. Seeing how stupidity and cruelty like witch- 
burning results from terror of the gods, he believed that human relations 
would become more wholesome as men learned to recognize the orderly 
routine of a world in which no divine dispensation gives them a natural right 
to keep slaves. So Aristotle's hatred of materialism receives a sufficient 
explanation from his partiality for slavery. 

The materialism of Democritus had two fundamental tenets. The first is 
what we now call the conservation of matter. It is expressed at the conclusion 
of an eloquent passage by the Epicurean poet Lucretius in the memorable 
words : 

When human life to view lay foully prostrate upon earth crushed down 
under the weight of religion, who shewed her head from the quarters of heaven 
with hideous aspect lowering upon mortals, a man of Greece ventured first to 
lift up his mortal eyes to her face and first to withstand her to her face. Him 
neither story of the gods nor thunderbolts nor heaven with threatening roar 
could quell, but only stirred up the more the eager courage of his soul, filling 
him with desire to be the first to burst the fast base of nature's portals. . . . 
On he passed far beyond the flaming walls of the world and traversed in mind 
and spirit the immeasurable universe; whence he returns a conqueror to tell 
us what can, what cannot come into being, in short on what principle each 
thing has its powers defined, its deep-set boundary marked. Therefore religion 
is put under foot and trampled upon in its turn. Us his victory brings level 
with heaven. This is what I fear herein, lest haply you should fancy that you 

N* 



394 Science for the Citizen 

are entering on unholy grounds of reason and treading the path of sin, whereas 
on the contrary heinous religion has given birth to sinful and unholy deeds. 
. . . You yourself some time or other overcome by the terror-seeking tales 
of the seers will seek to fall away from us. Ay, indeed, for how many dreams 
may they now imagine for you, sufficient to upset the calculations of life and 
trouble all fortunes with fear. And with good cause, for if men saw that there 
was a fixed limit to their woes, they would be able in some way to withstand 
the religious scruples and threatenings of the seers. . . . This terror then and 
darkness of mind must be dispelled not by the rays of the sun and glittering 
shafts of day, but by the aspect and the law of nature, whose first principle we 
shall begin by thus stating, nothing is ever gotten out of nothing by divine power. 
Fear in sooth takes such a hold of all mortals, because they see many operations 
go on in earth and heaven, the causes of which they can in no way understand, 
believing them therefore to be done by divine power. For these reasons when 
we shall have seen that nothing can be produced from nothing, we shall then 
more correctly ascertain that which we are pursuing, both the elements out 
of which everything can be produced and the manner in which all things are 
done without the hand of the gods. 

The vacuum pump destroyed this terror and darkness of mind. It estab- 
lished a new realm of matter, and removed the air we breathe from the domain 
of "spirits/* Lest we be accused of emphasizing the "material" benefits ot 
scientific progress unduly, we may recall the fact that witch-burning died out 
in England during the three decades which followed the revival of materialism. 
In Scotland, where belief in the authority of the seers was more firmly 
rooted, the amiable practice persisted longer. As late as 1664 one traveller 
records the spectacle of nine old women dying at the stake on the same day 
in Leith for dispensing herbs contrary to Scripture. When at length correct 
information about the way in which chemical changes can be brought about 
had become more generally available, the execution and torture* of people 
supposed to be capable of suspending the laws of nature came to an end. 

* The following citation from R. D. Melville in the Scottish Historical Review, 
vol. 2 (1905), tells of the ingenious sources of entertainment of which we have been 
robbed by the decline of Calvinism. 

"The barbarity of the tortures wreaked upon persons . . . suspected of witchcraft 
or sorcery is sufficiently instanced by the well-known case of ... Dr. Fian, alias 
Cunningham, schoolmaster at Saltpans, in Lothian. . . . In the first place, his head or 
neck was 'thrawn' or twisted with a rope, he was then 'put to the most Severe and 
Cruell paine in the world called the^bootes'; shortly afterwards the nails of all his 
ringers were torn out with pincers, two needles having previously been thrust under 
every nail 'over even up to the heads'; this proving unavailing to extort a confession, 
he was again subjected to the boot, 'wherein he did continue a long time, and did 
abide so many blows in them that his legges were crusht and beaten together as small 
as might bee; and the bones and the flesh so brused that the blood and marrow 
spouted forth in great abundance, whereby they were made unserviceable for ever. 5 
But continues the report 'all these grievous paines and cruel torments' failed to 
extort a confession, 'so deeply had the Devil entered into his heart/ Thereafter, 
by way of a terror and example to all others 'that shall attempt to deall in the lyke 
wicked and ungodlye actions as witchcraft, sorcerie, conspiration, and such like/ 
Fian was condemned to die in the special manner provided by the law of the land 
'on that behalfe'; and he was accordingly conveyed in a cart to the Castle Hill of 
Edinburgh, and having first been strangled at a stake, his body was thrown into a 
fire, 'ready provided, and there burned ... on a Saterdaie in the end of Januarie 
last past, 1591.' The narrative then quaintly but significantly proceeds to observe 
'The rest of the witches which are not yet executed, remayne in prison till farther 
triall and knowledge of his Majestie's pleasure.' " 



The Third State of Matter 395 

In Scotland the last cases occurred about 1727. Lecky tells us that nine years 
later the ministers of the associated presbytery met to record with grief their 
inability to stem the tide of modern doubt before turning their attention to 
new expedients for satisfying the deeper needs of mankind. 

In 1643, Sir Thomas Browne,, who was more enlightened than most of 
his contemporaries, wrote of witches : "they that doubt of these ... are 
obliquely and upon consequence a sort not of Infidels but Atheists ... the 
Devi hath them already in a Heresie as Capital as Witchcraft." His views 
on Lucretius were appropriate. "I do not much recommend the reading or 
studying of it, there being divers Impieties in it, and 'tis no credit to be 
punctually versed in it." Nevertheless the new naturalistic temper which 
Browne did much to sponsor was gaining ascendancy over what Browne 
himself called "the mortallest enemy unto knowledge and that which has 
done the greatest execution upon truth." The testimony of the surgeon was 
becoming as influential as the counsels of the Church. Needham relates a 
significant episode : 

On May 16, 1634 3 Sir William Pelham wrote to Lord Conway as follows: 
"The greatest news from the country is of a huge pack of witches which are 
lately discovered in Lancashire, wherof it is said 19 are condemned and at 
least 60 already discovered; there are divers of them of good ability, and they 
have done much harm. 'Tis suspected that they had a hand in raising the great 
storm wherein his Majesty was in so great danger at sea in Scotland." The 
women examined by Harvey and his colleagues were the survivors of a batch 
of seven, three of whom had died in prison, and they had already been examined 
by Bishop Bridgeman of Chester, who had made a hair-raising report on their 
spiritual condition. In due course Mr. Alexander Baker and Mr. William 
Clowes, both Surgeons to the King, Dr. William Harvey, his physician, with 
six other medical men and ten midwives co-opted by them, proceeded to the 
Ship Tavern at Greenwick, where the prisoners lay, in order to make their 
examination. On the report which was subsequently made Harvey's signature 
does not appear, but instead of it that of Alexander Reid, M.D., the Anatomy 
Lecturer at Surgeons' Hall. The upshot of the affair was that no physical sign 
of diabolical intercourse could be found on the bodies of any of the women, 
no abnormal teats, cutaneous discolorations, or satanic brandings. And the 
report was sent in to that effect, stating that the examination had been under 
the direction and in the presence of Dr. William Harvey. 

A second tenet of the Greek materialists was that matter is not continuous. 
Before there were good thermometers little was known about the expansion 
of solids or liquids. Compressibility was especially the characteristic of matter 
in the airy state. The recognition that air is material had to take account of 
this fact. The Greek materialists drew the bold conclusion that matter must 
be made up of weighty particles separated by a variable expanse of empty 
space. English physicists of the seventeenth century, engaged in studying the 
rules of gas compression, naturally welcomed the revival of the atomistic 
doctrine. A century later it furnished the clue to the fixed proportions in 
which elements combine to form new compounds. Seemingly the stage was 
now set for a great theoretical development of chemistry. This did not come 
till a much later date. Like those who now tell us that economics is one 



396 



Science for the Citizen 



subject and psychology is another, the professional chemists of the day were 
still following the trail of self-evident principles. 

ELASTICITY, COHESION, AND ADHESION 

How the existence of pulling power exerted by the particles of which 
matter is supposed to be made up may affect the behaviour of matter in bulk 
is illustrated by the phenomenon of elasticity or the resistance of solids to 




(a) 







(a.) 

FIG. 240. ADHESION AND COHESION 

The cohesive power of water is less than its adhesive power to glass. Hence water (a) 
clings to glass and wets it, (b) exhibits a concave meniscus, and (c) creeps up a fine 
tube. The cohesive power of mercury is greater than its adhesive power for glass. 
Hence mercury (a) does not cling to glass or wet it, (&) exhibits a convex meniscus 
and (c) sinks below the gross equilibrium level in a fine glass tube. The height of 
a convex meniscus is easier to read than that of a column which wets glass. Hence 
mercury is better for thermometers or barometers. (It is also better for thermometers 
because it expands regularly over a wide range, and has a high boiling point. It is also 
better for barometers because it is very dense and has a very low vapour pressure 
see p. 566.) 

change of shape. A large quantity of a liquid exhibits no analogous phe- 
nomenon. It takes the shape of the vessel in which it lies, and presents a flat 
level surface where it is free to move except in so far as the puU of terrestrial 
gravity holds it in position. When the behaviour of liquids in fine drops, 
thin films, and very narrow tubes is examined new characteristics manifest 



The Third State of Matter 397 

themselves. They were first studied in the same social context as the air 
pump and the water-wheel by an English clergyman, Stephen Hales, who 
published an account of experiments on the ascent of sap in 1727. 



MILKS 
SEA! 
15 

14 
13 
12 
11 


ABOVE flC 2 ^ ffinZHtD?) = = -3j"'Q'5/.S. 2 

- HVEL length, of stands pmduhim = 38- 93 in. 












30 

106 

125 

2OO 
250 

1 

37-5 

500 

625 
7<SO 






U.S. A .record, 1935. 


13-7nul^s (water boils at 29C) 


(waerbdl S l52C) 




9 


CJ Ptcarcl , 1931 . 9*6 miles 
** (water fcozis ai 57 &) 


7 




Cirrus cLotids (upper cloud limit) 


4 




/( 

/ 


Tj\Mt.Evcrcst 
\ 


^ 


a 


Us s(t /^-C) 


3 


A 
, A_ _j 


n\ -^ 

1 \~ d 


Blanc (wazr bolla at SyC) 
]\- Fr^lngby. d*QL WJ^i- 


1 


1 


(11 C^ 
ibis iSdi 

II III 


Istr'acfcos cbu 
-...I1L ! Cr'A 1 !l 7'r 


l\ K 


It" 

1 


^v BfflTL ISfevtSs fooiLS 


^iWl 


1 


|\ ytf]Trr7\J^ 95c 



g (at Equates) == 32- O<? ft. sec 2 
length of seconds pmdultuTi = 3.9 -01 iti. 





of 



water IQOC 



FIG. 241. THE EARTH'S ATMOSPHERE 

One of these characteristics, called cohesion, is seen when a soap film is 
stretched,, or drops of oil are allowed to trickle on to a large surface of water. 
The other, called adhesion, is seen when a film of water clings to the finger 



398 Science for the Citizen 

after it is withdrawn from water in which it was previously immersed. 
According to the Newtonian view of matter, cohesion is interpreted as the 
result of the pull which particles of the same substance exert on one another. 
The result of this is that whereas all the particles in the centre of a liquid 
are pulled equally in all directions, the particles at the surface can only 
pull on one another and those below them, and behave like a stretched sheath. 
Hence the surface particles are only in equilibrium when they are all arranged 
symmetrically around the central bulk. If this is small, so that the total 
pulling power of gravity on the bulk of the fluid is also small, the liquid 
assumes the spherical form of a drop. Adhesion, on the other hand, repre- 
sents the pulling power of particles of one substance on those of another. 
When this is large compared with the force of gravity, as when the bulk of 
the water has trickled off the finger, it may be sufficient to resist further 
separation beyond a certain point. A film of fluid therefore clings to the 
surface with which it is in contact and "wets" it. Some fluids exhibit very 
little adhesion for particular surfaces, and do not "wet" them. 

In general, the behaviour of a liquid at its surface of contact with a solid 
is determined by the interplay of both these characteristics. If the fluid does 
not wet the solid, i.e. if the power of adhesion is relatively insignificant as 
compared with cohesion, the surface of the liquid tends to assume the 
spherical shape so that the edge of the liquid in contact with the solid is 
depressed (Fig. 240), and the meniscus or air-liquid surface is convex 
upwards. If the fluid does wet the solid the edge of the fluid in contact with 
it is drawn upwards, and the meniscus is concave upwards. The result is 
that in tubes of very fine bore the rule that liquids find their own level is no 
longer true. If the liquid (e.g. mercury) does not wet a glass tube, its surface 
is depressed appreciably below the gravity level. If (e.g. water) it does wet 
the glass, it rises appreciably above it. In exceedingly fine tubes, like the 
minute vessels in the wood of plants, or the narrow spaces between particles 
of soil, water may rise to a very considerable height on this account. 
Capillarity, as the ascent of fluids in very fine tubes is called, can be illustrated 
by substituting a piece of dead bamboo for the wick of a paraffin lamp. The 
paraffin creeps up the bores, and the bamboo acts as an efficient wick. In an 
ordinary oil-lamp or candle the wick itself acts in the same way. The oil or 
molten wax creeps up to the flame through the fine spaces between the 
threads. 

THINGS TO MEMORIZE 

1. The initial downward acceleration of a body of density D in a fluid of 

/ d\ 
density d is given by a = ( 1 - Jg. 

2. Boyle's Law. pv = k, or, where d is the density p = Kd and d = K'p. 



The Third State of Matter 399 



EXAMPLES ON CHAPTER VII 

1. In the following table the sign indicates that the substance sinks, 
4- indicates that it floats and that it does not decidedly do either. Find the 
approximate density of each of the solids. 



Liquid 


Density 
(gm. 
per c.c.) 


Larch- 
wood 


Butter 


Beeswax 


Resin 


Glue 


Celluloid 


Gasolene 


0-69 


+ 


_ 


._ 




__ 


_ 


Ether.. 


0-74 


-f 





-_ 











Ethyl alcohol 


0-70 


4- 













__ 


Wood spirit . 


0-81 


-f 











_ 





Turpentine 


0-87 


-f 

















Olive oil 


0-92 


+ 


+ 


_ 








_ 


Castor oil 


0-97 


-f 


+ 













Water . . 


1 - 00 


-f 


-f- 


-f 











Creosote 


1-10 


i 


+ 


+ 











Glycerine 


1-26 


+ 


-f 


+ 


+ 








Chloroform . 


1-48 


f 


+ 


+ 


+ 


+ 


+ 



2. Taking the density of mercury as 13-5 gm. per c.c., iron 7-8, gold 19 -3, 
lead 11-35, silver 10-5, aluminium 2-7, and magnesium 1-74, calculate the 
acceleration with which each of these metals will begin to rise or fall under 
gravity when placed in (a) mercury, (6) chloroform., (c) water, (d} gasolene. 

3. If the density of sea water is 1-025 gm. per c.c., what will be the pressure 
in kg. per sq. cm. at a depth of 1 km. below the surface of the sea, neglecting 
atmospheric pressure? 

4. A U-tube of uniform bore of diameter 1 cm. is about half filled with water. 
If the density of oil is 85 gm. per c.c., how many c.c. of oil must be poured 
into one limb to make the level of the water rise 6 cm. in the other? 

5. At what angle to the vertical must a glass tube open at the top and con- 
taining water be gradually tilted until the pressure at the foot of it due to the 
water is diminished by half? 

6. What is the ratio of the pressure on the upper half of one side of a cistern 
full of water to the pressure on its lower half? 

7. The lock gate of a canal is 14 feet wide and 9 feet deep. How great is the 
pressure in tons weight on one side of it, if the water is level with the top and 
the density of water is 62 4 lb. per cub. ft. ? 

8. In a hydraulic press the cross-sectional areas of the small and large 
plungers are respectively 1 sq. in. and 100 sq. in. If the mechanical advantage 
of the lever handle is 5, what is the upward force of the ram when the lever 
handle exerts a force of 1 cwt.? 

9. If the densities of glycerine, water, and ethyl alcohol, are respectively 1-26, 
1-0, 0-79 gm. per c.c., find the initial acceleration of a piece of iron of density 
7-86 gm. per c.c. falling in each. 

10. The diameter of a small plunger in a hydraulic press is 1 inch and that 
of the large plunger 1 foot. What is the upward force exerted by the ram 
when a downward force of 56 lb. is applied to the smaller plunger? If the smaller 
piston moves through 1 J inches in one stroke, find how far the ram has moved 
after 10 strokes. 

1 1 . Find the height of a barometer in which oil is used of density 0-845 gm. 



400 Science for the Citizen 

per c.c., knowing that the height of a mercury barometer is 752 mm. and the 
density of mercury is 13-6 gm. per c.c. 

12. A closed vertical tube has its open end inserted in a bowl of water. It is 
noticed that the surface of the water is 10 feet below the closed end of the 
tube. Knowing that the height of the water barometer is 33 feet, what is the 
pressure in the enclosed space due to the column of water? 

13. Observation shows that the barometer falls an inch for every 900 feet 
above sea-level. A man takes a pocket aneroid barometer, which is used to 
determine the heights of mountains, up a mountain, ascending in 3 hours and 
descending in 1J hours. The barometer registers 29-0 inches at the foot, 26-8 
inches at the top, and 28-7 inches at the foot of the mountain after descending. 
Estimate the height of the mountain. 

14. What is the atmospheric pressure in Ib. weight on a circular disc 
3 inches in diameter? 

15. A man blowing into one limb of a U-tube half filled with mercury causes 
a difference of level of 4 cm. in the two limbs. What pressure does he exert in 
dynes per sq. cm.? 

1C. What is the volume of a quantity of gas at standard pressure, i.e. 76 cm., 
if, when the barometer reads 72 cm., the volume of the gas is 159 c.c.? Find the 
pressure at which it would have a volume of 200 c.c. 

17. A cylinder 12 inches long has a piston halfway down so that there are 
equal quantities of air on each side of it. Find the ratio of the pressures on the 
two sides if the piston is pushed down so that it is only 1 inch from one end. 

18. A closed tube 150 cm. long is half filled with mercury, and its open end 
inserted in a bath of mercury, making sure no air escapes from the tube. If the 
barometer reads 75 cm., what is the length of the mercury column in the 
tube? 

19. A depression of the mercury column in a barometer is caused by a little 
air in the space at the top of the column, so that the barometer reads 28 inches 
instead of 29 inches, and 29 inches instead of 30-2 inches. Determine the 
correct value of the atmospheric pressure when the erroneous reading is 
29-4 inches. 

20. A thin tube of soft glass is taken and a thread of mercury 8 cm. long is 
drawn into it. One end of the tube is sealed off in a bunsen flame entrapping 
a quantity of air between the mercury and the end. When the tube is suspended 
with its closed end down, the length of the air column is 16-1 cm., but when 
it is suspended with its closed end up, the length of the column of air is 20 cm. 
What is the pressure of the atmosphere? 

21. If the mercury barometer reads 45 inches inside a cylindrical diving 
bell 8 feet high, of which the top is submerged 12 feet below the surface of the 
water, how high has the water risen in the bell? 

22. If a litre (1,000 c.c.) of dry air at 76 cm. is found to weigh 1 3 gm., how 
much space does 2 gm. of dry air occupy at the same temperature when the 
pressure is 70 cm.? 

23. A closed cylinder of sectional area 10 sq. cm. stands 25 cm. above the 
level of water in a wide dish. The level of water in the cylinder is 20 cm. above 
the level of water in the dish. Find (a) the pressure of the air in the cylinder in 
terms of the mercury barometer, taking the specific gravity of mercury as 13; 
(&) how high the cylinder will stand above the level of water in the dish, if it 
is depressed till the level of fluid inside and outside is the same. 

24. A rubber balloon weighs 5 gm. With what acceleration will it begin to 
fall if it is blown up till its cubic capacity is (a) 750 c.c., (b) I litre, (c) 2 5 litres? 



The Third State of Matter 401 

Assume the pressure inside the balloon to be 1 J atmospheres in each case, and 
that the density of air is 0-0013 gm. per c.c. 

25. When a U-tube hydrometer as shown in Fig. 239 is inverted with one 
limb in mercury and the other in water, the fluids rise respectively to 0-75 cm. 
and 9-75 cm. when some air is sucked out. A little air is readmitted, and the 
mercury drops to 0-4 cm. What is the water level? 

26. One limb of the same apparatus dips in mercury, the other in turpentine. 
The level of the turpentine is 8 9 cm. and that of the mercury is 6 cm. Use 
the data of the last example to find the height of the water column if one limb 
dipped in water and the level of turpentine in the other was 1 cm. 

27. The glass tube of a mercury barometer of sectional area 0-75 sq. in. 
stands 38 inches above the level of mercury in a large deep dish. The level of 
fluid in the latter does not vary appreciably when all the fluid in the tube is 
removed. If the atmospheric pressure is 30 inches, how much will the mercury 
be depressed by introducing a bubble of air (a) 0-001, (b) 01, (c) 1 cub. in.? 

28. If at a certain temperature 1 gm. of air occupies 1 litre, at 76 cm. pressure 
on the mercury barometer, we can put instead ofpv = k 

76 x 1,000 = k. 

At that temperature k has the value 76,000 when the unit of pressure is 1 cm. 
of mercury and the unit of volume is 1 c.c. Similarly calculate for the same 
temperature the numerical value of k (a) when the unit of volume is the litre 
and the unit of pressure the "atmo," i.e. the pressure exerted by a column of 
mercury 76 cm. in length; (&) when the unit of volume is the cubic centimetre 
and the unit of pressure is 1 dyne per sq. cm. (Density of mercury = 13-6 gm. 
per c.c., and# = 981 cm. per second per second.) 

29. At C. 1 gm. of hydrogen was found to occupy 11-2 litres at 760 mm. 
on the mercury barometer. At the same temperature 1 gm. of oxygen occupied 
0-710 litres when the pressure was 750 mm., and 1 gm. of nitrogen occupied 
0-789 litres at 770 mm. Give the density in gm. per c.c. of the three gases, and 
express to the nearest whole number the relative densities of oxygen and nitrogen 
taking hydrogen (RD = 1) as standard. 

30. The density of hydrogen at C. and 760 mm. pressure is 0-00009 gm. 
per c.c. The relative densities of ammonia, carbon dioxide, and sulphur dioxide, 
are respectively 8 5, 22, and 32. Find how much space at C. 1 gm. of ammonia 
would occupy at 750 mm., 1 gm. of carbon dioxide at 740 mm. and 1 gm. of 
sulphur dioxide at 770 mm. 

31. The density of petrol is about 0-7 gm. per c.c. and of cork 0-25. If air 
at C. and 760 mm. pressure is 14 J times and sulphur trioxide is 40 times 
as dense as hydrogen, compare the initial acceleration under gravity of a drop 
of mercury, a rain drop, a drop of petrol, and a piece of cork, falling in each of 
these three gases at a place where g in vacuo is 32 ft. sec. 2 Use any relevant 
data in the foregoing examples. 

32. At a place where the value of g is 32 ft. sec. 2 , compare the period of a 
platinum pendulum which beats seconds in vacuo, if made to swing in an 
atmosphere of (a) air, (fc) hydrogen, (c) sulphur trioxide. Neglect atmospheric 
resistance. 



CHAPTER VIII 

THE REBIRTH OF MATERIALISM 

The Exhaustion of the Neolithic Economy 

PHARMACEUTICAL establishments are sometimes called chemist's shops. Until 
the Industrial Revolution what chemistry was taught in the official seats of 
learning was a branch of medicine. According to a statement of Voltaire in 
the Dictionnaire Philosophique, the Parliament of Paris in 1624 passed a law 
which compelled the chemists of the Sorbonne to conform to the teaching 
of Aristotle on pain of death and confiscation of goods. The great intellectual 
progress which occurred in the nascent Protestant democracies during the 
ensuing half century is illustrated by the words of Joseph Glanvill, who is 
notable for his tract on witch-burning. In his book Modern Improvements of 
Useful Knowledge published in 1675, Glanvill wrote: 

Among the Egyptians and Arabians, the Paracelsians and some other moderns 
chymistry was very phantastic, unintelligible and elusive. . . . But its later 
cultivators, and particularly the Royal Society, have refined it from its dross and 
made it honest and sober, and intelligible, an excellent interpreter to philosophy 
and help to common life. For they have laid aside the delusory designs and vain 
transmutations, the Rosicrucian vapours, magical charms and superstitions, 
and formed it into an instrument to know the depths and efficacies of nature. 
And this is no small advantage that we have above the old philosophers of the 
Notional way. 

The Notional way was the phrase which Sprat (p. 552) and the founders 
of the Royal Society referred to the Aristotelian attempt to found a science 
on self-evident principles. Events proved that chemists, like the economists 
of our own time, were reluctant to abandon it. In this chapter we shall 
see how new social needs which accompanied the industrial expansion of 
the seventeenth and eighteenth centuries refined chemistry from the dross 
of Aristotelian philosophy and the Rosicrucian vapours to which Joseph 
Glanvill alludes. 

Mercury combines with the yellow gas chlorine to form two different 
substances. One is the highly poisonous corrosive sublimate (mercuric 
chloride), now used as an antiseptic for washing wounds. The other, which 
contains twice the ratio of mercury to chlorine by weight, is the purgative 
calomel (mercurous chloride). In an old chemical work, written before the 
recognition that air has weight, the formation of the calomel by grinding 
crystals of corrosive sublimate with the liquid metal itself was described by 
the statement, "the fierce serpent is tamed, and the dragon so reduced to 
subjection as to oblige him to devour his own tail." The words will not 
appear to be pure rhetoric, if we recall the expedients employed to make the 
supposedly imponderable spirits pass out of the air into the ingredients of 
the retort. Ancient number magic had associated the seven best-known 
metals with the seven bodies classified as planets in the pre-Copernican 



The Rebirth of Materialism 403 

astronomy: the Sun with gold, the Moon with silver. Mercury with mercury, 
Venus with copper. Mars with iron, Jupiter with tin, and Saturn with lead. 
Success in conjuring the spirits which presided over chemical processes in 
which these metals formed compounds, or were extracted from their com- 
pounds, was not to be expected unless the experiment was carried out when 
the appropriate celestial body was making its transit above the meridian. 
The various conjunctions of the several planets offered specially propitious 
circumstances for carrying out reactions in which different metals participated. 
Only three centuries have passed since these beliefs were widely accepted. 



==- ~=f^tfls=? 




FIG 242. AN OLD SYMBOL REPRESENTING THE DISSOLUTION OF GOLD 
IN AQUA REGIA 

We are apt to dismiss them as magic, and to forget that the distinction 
between the world of matter and the world of spirit was a very tangible one 
when theology upheld witch-burning. In the sixteenth century its professors 
had not conveniently fixed the boundary where the realm of possible proof 
and disproof ends. Medieval Europe accepted the authority of Aristotle 
and of the Apostle Paul. According to the Pauline view stated in the First 
Epistle to the Corinthians (1 Cor. xv. 40-52), anything which did not obey 
the Aristotelian law of gravitation was ipso facto spirit. The resurrected body 
was not invisible. It was merely imponderable, as the vapours of the alchem- 
ical retort were then supposed to be. A literal belief in the resurrection offered 
no difficulties in an age when the basic processes of medicine and metallurgy 
were believed to be modelled on the same plan. 

The discovery that air has weight, and that air is, in short, a form of 
ponderable matter, carried with it the recognition that there is a third state 
of matter in contradistinction to the liquid and the solid state. Since there 
are many different sorts of matter called solids or liquids, it was natural to 
suppose that other substances may share with air the characteristics which 



404 Science for the Citizen 

distinguish air from matter in the liquid or the solid state. A new word 
gas now replaced "spirits," derived from an equivalent Teutonic word which 
did not carry into international usage the mystical content of its original 
meaning. It was first used by van Helmont in 1648, about the time when 
Torricelli's experiments gave new and conclusive evidence for the weight 
of the air. During the century and a half which elapsed between the death 
of van Helmont and the beginning of the nineteenth century the arts of 
manufacture received a new impetus from discoveries which made it possible 
to distinguish different forms of air, or, as we now say, different gases. 
These discoveries led to new rules about how substances combine or can 
be broken down to form new ones. What we call modern chemistry, in contra- 
distinction to the chemistry of antiquity or the Middle Ages, is the theory 
of manufacture based on the knowledge that different gaseous elements enter 
into the composition of different objects of use. 

The recognition of the gaseous elements bears a very close relation to the 
growth of mining and its problems in the sixteenth century. Agricola's De 
Re Metallica (1566) gives us a comprehensive picture of the class of technical 
problems which stimulated research into the physical properties of air. 
The story it tells is very different from the description of mining given by 
classical historians like Diodorus Siculus. The miner is no longer a slave in 
a chain-gang working under the overseer's' lash. "A miner, since we think 
he ought to be a good and serious man," writes Agricola, "should not make 
use of an enchanted twig, because if he is prudent and skilled in the nature 
of signs, he understands that a forked stick is of no use to him." The sixth 
book of Agricola's treatise, dealing especially with the problems of pumping 
and ventilation (Fig. 243), ends with a section on "the ailments and accidents 
of miners, and the methods by which they can guard against these." It is 
not an exaggeration to say that these represent the two principal themes 
which underlie the recognition of the individuality of gases. 

With the growth of deep-shaft mining for metals and the introduction of 
coal as fuel in the latter part of the sixteenth century two new problems had 
emerged in the everyday life of mankind. One was : What makes air foul to 
breathe or capable of sustaining life? The other was: What makes air inflam- 
mable and explosive like gunpowder, or incapable of supporting combustion? 
The phenomenon of combustion itself took on a new complexion through 
the introduction of gunpowder and the use of coal for fuel. Previous sources 
of fuel had been exclusively animal (e.g. tallow) or vegetable (wood and 
charcoal). That a seemingly mineral substance of "earthy" origin should 
produce fire was contrary to the prevailing belief that all substances were 
built up by combinations of earth, air, fire, and water, or of sulphur, salt, 
and mercury. The introduction of steam power, beginning with such inven- 
tions as the Marquis of Worcester's patent at the end of the seventeenth 
century, made the nature of combustion a topic of increasing interest. Mean- 
while, the rapid industrial development which accompanied the successive 
introduction of water power and steam power was leading to the gradual 
exhaustion of the sources from which the crude chemicals employed by the 
traditional methods of the cloth, mining, glass, and soap-making industries 
were derived. 



The Rebirth of Materialism 405 





FIG. 243 

Two prints from Agricola's Treatise, showing the importance of ventilation in deep- 
shaft mining. The lower displays a pivoted barrel over the flue. The upper shows a 
ventilating fan placed above the flue and driven by a windmill. 



406 Science for the Citizen 

One important facet of the social background of the growth of chemical 
science in the seventeenth century is that warfare was making an increasing 
demand on gunpowder, and the success of the rising Protestant democracies 
depended on exploiting the new technique of self-defence and imperialist 
expansion. Of the three most usual constituents of "gunpowder" charcoal, 
sulphur, and saltpetre, i.e. nitrates the supply of the first from wood and the 
second from volcanic sources, offered comparatively little difficulty. The 
detection of sources of natural nitrates involved more analytical sophistication. 
Concerning the discovery of gunpowder, various dates are cited. The truth 
is that no precise date can be given. The various recipes which are called by 
that name emerged gradually from the practice of incendiary warfare before 
the means of purifying the constituents as yet existed. The use of burning 
pitch and oil, camphor and resins, to discomfit a besieging force is of great 
antiquity, and the certainty that explosive mixtures were known to the 
Chinese and in India before the use of artillery in Europe helps us to under- 
stand how the knowledge grew by easy stages. In these countries natural 
deposits of nitrates are formed where there are manure heaps. The accidental 
recognition that pitch contaminated with this natural "salt" would burn 
more vigorously cannot have been a specially remarkable discovery. 

The natural salt was first known as "Chinese Snow," later as saltpetre. 
Crackers and rockets made of bamboo packed with saltpetre and combus- 
tible material were used as incendiary devices before the construction of a 
metal cannon to propel a dart or ball was accomplished in the thirteenth 
century. The earliest recipes of explosive or incendiary mixtures include, 
in addition to saltpetre, resin and brimstone (sulphur), sulphur and charcoal, 
or sulphur, charcoal and camphor as the combustible constituents. Once 
the metal case was introduced, the destructive possibilities of gunpowder 
became a dominant feature of military technique, and a new industry which 
promoted the search for materials of dependable purity came into being. 
Marshall (Explosives, vol. 1) tells us that: 

In the fourteenth century gunpowder was only used on a small scale, and 
was made in ordinary houses with pestle and mortar. We hear, for instance, 
that the Rathaus at Lubeck was destroyed by fire in 1360 through the careless- 
ness of powder makers. Berthelot has stated that there were powder mills at 
Augsburg in 1340, at Spandau in 1344, and Liegnitz in 1348, but Feldhaus 
could find no confirmation of these statements in the archives of these towns. 
There is no mention of gunpowder or fire-arms in Augsburg before 1372 to 
1373, and the first powder mill was erected at Spandau in 1578. The scale of 
operations gradually increased, and in 1461 we find the first mention of a 
"powder-house" in the Tower of London; powder was made there for many 
years, as also in Porchester Castle. In the sixteenth century mills of considerable 
size were in existence: the Liebfrauenkirche in Liegnitz suffered at this time 
from the effects of explosions in a mill near by. In 1554 to 1555 a gunpowder 
mill is said to have been erected at Rotherhithe, and about 1561 George Evelyn, 
the grandfather of John Evelyn, the diarist, had mills at Long Ditton and 
Godstone, having learned the methods of manufacture in Flanders. A few 
years later he obtained from Queen Elizabeth a monopoly of the manufacture 
of gunpowder, which he and his sons were able to maintain more or less until 
1636, when Samuel Cordewell obtained the monopoly, which was abolished 



The Rebirth of Materialism 407 

by Parliament in 1641 3 the year before the outbreak of the Civil War. George 
Evelyn made a fortune out of gunpowder. 

Among the first papers communicated to the English Royal Society we 
find an account of the "History of the Making of Salt-Peter/' in which the 
author, Mr. Henshaw, declares: "... the only place therefore where salt- 
peter is to be found in these northern countries is in stables, pigeon houses. 
. . ." Henshaw gave a recipe for making "salt-peter" from the natural 
nitrate content of fertile earth. About 1650 the German chemist Glauber 
showed that saltpetre can replace manure as a means of restoring the fertility 
of exhausted soils . He called his book The Prosperity of Germany. The dis- 
covery of new sources of materials had now become a necessary prerequisite 
of prosperity. How Glauber's discovery that saltpetre can be used as a substi- 
tute for manure arose from the social practice of his times is illustrated well 
enough by the following passage from Marshall's treatise on the history of 
Explosives: 

In Europe there are very few localities where nitrate can accumulate in the 
soil to such an extent that a profit could be made by extracting it. There is no 
prolonged dry season during which deposits can form without being washed 
away again. Consequently saltpetre could only accumulate in sheltered places, 
such as cellars and stables, especially those in which there was much nitro- 
genous matter undergoing decomposition. As it was of the utmost importance 
in every country to have a sufficient supply of saltpetre, especially in time of 
war, its production formed the subject of royal decrees and orders at an early 
date. In France, officers (salpetriers commissiones) were appointed in 1540 to 
search for and extract saltpetre, and no doubt the industry was in existence 
some time before. This edict was confirmed and renewed in 1572, and again 
whenever France was waging a serious war. The saltpetre workers operated on 
the earth of stables, sheep-pens, cattle-sheds, cellars and pigeon-houses, and 
on the plaster and rubbish removed when houses were pulled down. They 
had the right to gather material everywhere, with scrapers and brushes in 
the houses, with picks and shovels in places not inhabited. No building or 
wall could be pulled down until notice had been given to the saltpetre workers, 
who stated which parts they wanted reserved. ... In the reign of Louis XIII 
(1610 to 1643) the annual crop of saltpetre amounted to 3,500,000 lb., but it 
gradually diminished in the eighteenth century largely on account of the strong 
objection the people naturally had to the presence of saltpetre workers in their 
houses and domains. . . . Until the sixteenth century saltpetre seems mostly 
to have been imported into England, much of it coming from Spain, but in 
1515 Hans Wolf, a foreigner, was appointed to be one of the King's gunpowder 
makers in the Tower of London and elsewhere. He was to go from shire to 
shire to find a place where there is stuff to make saltpetre of, and "where he 
and his labourers shall labour, dig or break in any ground." He is to make 
compensation to its owners. And in 1531 Thomas & Lee, one of the King's 
gunners, was appointed principal searcher and maker of gunpowder . . . 
gunpowder was only manufactured in England on a small scale until the second 
half of the sixteenth century when George Evelyn started mills on a compara- 
tively large scale. Consequently there was little difficulty before that time in 
obtaining sufficient saltpetre, but then it became necessary to grant the 
saltpetre men special privileges for digging up the floors of stables, dovecots 
and even private dwellings, and the kingdom was divided into a number of 



408 Science for the Citizen 

areas in which the collection and working of the saltpetre was assigned to various 
people. In 1561 Queen Elizabeth granted Gerard Honrick, a Dutchman, 500 
(or 300) for teaching two of her subjects how to make saltpetre. In 1588 she 
granted a monopoly for gathering and working saltpetre to George Evelyn, 
Richard Hills, and John Evelyn. The monopoly extended over the whole of 
the south of England and the Midlands, except the City of London and two 
miles outside it. In 1596 Robert Evelyn acquired the rights in London and 
Westminster from the licensees there. As a rule, however, the Evelyns did 
not work saltpetre themselves, but bought it from the saltpetre men. In the 
reign of Charles I there was considerable friction between the saltpetre men 
and the public, but it was probably due more to the weakness of the Crown 
than to any real difficulty in obtaining in England the quantity of saltpetre 
required, viz. 240 lasts per annum. There was also competition between the 
saltpetre men and the soap-boilers for wood ashes, which were then practically 
the only source of potash and were required for the conversion of sodium 
nitrate into the potassium compound. . . . The East India Company, then in 
its infancy, imported Indian saltpetre into England as early as 1625, and set up 
a powder mill in Windsor Forest, which, however, was stopped on the ground 
that it interfered with the King's deer. Next year the Company received a 
license to erect mills in Surrey, Kent, and Sussex. At this time its importations 
were on a small scale, but when its charter was renewed in 1693 it was stipu- 
lated that 500 tons of saltpetre should be supplied every year to the Ordnance. 
Ever since then, Indian saltpetre has been used very largely in England for the 
manufacture of gunpowder. 

The need for saltpetre affected chemical enquiry in two ways. It provided 
a direct incentive to studying the natural history of salts, and indirectly 
encouraged research into the way in which the green plant gets its food. 
Glauber's work, which showed that the saltpetre, i.e. nitrate content, of 
manure is mainly responsible for its effect on the fertility of the soil, stimu- 
lated parallel enquiries in England. Such experiments on soil fertility are 
among the first examples of chemical analysis, in which the weight of all 
the constituents of a reaction is tested. The same care in measuring the 
weight of substances taking part in a reaction is also shown in experiments 
suggested by the use of metallic alloys in making coins of the realm. An 
account of such experiments by Lord Brouncker is given in Sprat's History 
of the Royal Society. The paper entitled "Experiments on the Weight of 
Bodies increased in the Fire, Made at the Tower," gives examples of metals 
which gain weight when heated in air. 

Sprat's History also mentions that the Royal Society had encouraged "the 
chymical examination of French and English wines." The "Proposal for 
Making Wine" put forward by Dr. Goddard, and included in Sprat's 
collection of communications made to the Society in the first few years of 
its work, contains a passage worth quoting : 

It is recommended to the Care of some skilful Planters in Barbadoes, to try 
whether good Wine may not be made out of the Juice of Sugar-canes. That 
which may induce them to believe this Work to be possible, is this Observa- 
tion, that the Juice of Wine, when it is dried, does always granulate into Sugar, 
as appears in Raisins, or dried Grapes : and also that in these Vessels wherein 
a cute, or unfermented Wine is put, the Sides are wont to be cover'd over with 



The Rebirth of Materialism 409 

a Crust of Sugar. Hence it may be gather'd, that there is so great a Likeness 
of the Liquor of the Cane, to that of the Vine, that it may probably be brought 
to serve for the same Uses. If this Attempt shall succeed, the Advantages of 
it will be very considerable. For the English being the chief Masters of the 
Sugar Trade, and that falling very much in its Price of late Years, while all 
other outlandish Productions are risen in their Value; it would be a great 
Benefit to this Kingdom, as well as to our Western Plantations, if Part of our 
Sugar, which is now in a manner a meer Drug, might be turn'd into Wine, 
which is a foreign Commodity, and grows every Day dearer; especially seeing 
this might be done, by only bruising and pressing the Canes, which would 
be a far less Labour and Charge, than the Way by which Sugar is now made. 

The subsequent history of enquiries on these lines justified itself by the 
promotion of one of the earliest industries based on direct application of 
chemical knowledge. The beet-sugar industry was destined to play a most 
important part in the beginnings of what we now call "organic chemistry." 
Cohen tells us that 

the presence of sugar in beetroot was observed in 1747 by the German chemist 
Marggraf who suggested the cultivation of beet as a source of sugar, but the 
early attempts to utilize it commercially proved unprofitable. The success 
of the industry dates from about the year 1830 when important improvements 
began to be introduced. Careful selection of seed and improved cultivation 
nearly doubled the quantity of sugar in the beet. The use of steam-heated 
vacuum pans gave a larger yield of crystallizable sugar and new mechanical 
appliances for saving labour lowered the cost of production. 

Another industry which played an important part in creating the demand 
for chemical knowledge was the manufacture of earthenware and china. In 
England it did so indirectly by stimulating the demand for coal, as well as 
by its immediate need for information about the nature of impurities in 
local clays and marls (brickearths), the art of colouring the finished product, 
and glazing. Dr. Plot, one of the earliest secretaries of the Royal Society 
and a Professor of Chemistry at Oxford, in his book The Natural History of 
Staffordshire (1686) tells us "for making the severall sorts of pots they have 
as many different sorts of clay ... the best being found near the coale." Thus 
the juxtaposition of coal and boulder clay seams on the site of an industry 
which made heavy demands on fuel for baking, made the Potteries a focal 
centre of industrial development in the eighteenth century. Even in the 
seventeenth century it called for chemical skill. The colours, says Plot, were 
obtained by using different varieties of "slip" clays "except the motley 
colour which is procured by blending lead with manganese, by the workmen 
called magnus" In a book on the history of the same county, published in 
1798, Shaw tells us that in the eighteenth century the natural clays of the 
neighbourhood were little used, because they were metalliferous : 

each having a portion of oxide of iron ... the clays from Dorset or Devon 
have all their impurities extracted before they are vended to the purchasers 
. . . being extremely white when fired owing to being scarcely impregnated 
with oxide of iron, which would make the ware yellow or red in proportion to 
the quantity of clay. 



4io Science for the Citizen 

Wedgwood, the "Prince of Potters," was in the forefront of the cultural 
renaissance in the latter half of the eighteenth century. As an industrial 
leader, he founded the first chamber of commerce in England. As an experi- 
menter, he was elected a Fellow of the Royal Society on his own merits. His 
attitude to the relations of industry and science comes out in a letter to 
Bentley (1767). 

I am going on with my experiments upon various earths . . . many of my 
experiments turn out to my wishes and convince me more and more of the 
extensive capability of our manufactures for further improvement . . . such a 
revolution is at hand and you must assist in and profit by it. 

As with coal mining, the human problem of the Potteries had repercus- 
sions in the domain of chemistry. At the end of the eighteenth century 
interest in occupational diseases (see Chapter XVI) had materialized in a 
lively public health agitation, of which one of the most notable figures was 
Dr. Percival, of Manchester. Percival and Dr. Gouldsdon, of Liverpool, 
vigorously exposed the dangers of lead poisoning in the glazing of Pottery 
products. Wedgwood was much embarrassed by their pamphlets. We find 
him writing to Bentley (about 1775), "I will try in earnest to make a glaze 
without lead, and if I succeed will certainly advertise it." Leadless glazes, 
of which the ingredients were either lime, alkaline carbonates ("potashes"), 
or borates (imported from Tuscany), were introduced in the latter end of the 
eighteenth century under the impact of the public feeling aroused. 

The new sense of social responsibility which emerges in the medical 
profession in the latter half of the eighteenth century also revived interest 
in the problem of ventilation which had arisen earlier in connection with 
deep-shaft mining for coal and tin. Davy, who seems to have acquired his 
interest in theoretical chemistry from an early friendship with the younger 
son of James Watt, the engineer, obtained his first scientific employment in 
the "Medical Pneumatic Institution" at Clifton, Bristol. This was founded 
by Beddoes, who partly on account of his strong Jacobin views had left 
the chair in chemistry at Oxford to take up medical practice. Davy's first 
important chemical researches were concerned with laughing gas (nitrous 
oxide), and one of his great inventions was the miner's safety lamp. 

RESPIRATION 

Among the "information they have given to others to provoke them to 
enquire," Sprat mentions that the founders of the Royal Society included 
in their programme "the fatal Effects of Damps on Miners and the Ways 
of recovering them." The human aspect of ventilation in mines was not 
the only practical issue which provoked enquiry into respiration. In the 
same context as "Relations ... of deep Mines and deep Wells" Sprat also 
mentions 

Relations ... of Divers and Diving, their Habit, their long holding of breath 
and of other notable Things observed by them. 

In the social context of early coal mining the discovery that air is a form 
of matter was highly propitious to a systematic attack on the subject-matter 



The Rebirth of Materialism 



411 



of Agricola's book dealing with "health and accidents of miners." The im- 
portance of this theme is borne out by the fact that "pneumatic" chemistry 
was a recognized major branch of the subject during the eighteenth century. 
During the Middle Ages breathing had been a spiritual accomplishment. 
The identification of "spirits" with the gaseous state of matter is implied 

Vw tln* cxmrvnvmnnc IICP nf tV T atin 'word animn. for soul and breath. So% 





FIG. 244. THE VALVES IN THE VEINS 

On the left the figure shown by Fabricius. On the right diagrammatic view of 
three sets of valves. 

too, the Hebrew scriptures declare that man became a living spirit when 
the Deity breathed into his nostrils the breath of life. Such assertions offered 
no difficulties to the literate classes of Europe while they continued to share 
Aristotle's contempt for slave labour. Modern doubt begins with the study 
of the common pump. 

Precise information about what breathing really involves came through 
the recognition that the heart performs the part of a pump maintaining a 
continuous circulation of blood from the lungs to the tissues, and the tissues 
to the lungs. In the latter half of the sixteenth century Servetus had con- 
cluded that all the blood from the right side of the heart is pumped through 
the minute vessels of the lungs and returns thence to the left. Owing to a 
difference in celestial arithmetic between himself and Calvin, who had lately 
established the kingdom of God in Geneva, Servetus was burned at the 
stake before he had time to complete his researches. Half a century elapsed 



412 



Science for the Citizen 



before the English physician Harvey convinced his contemporaries that the 
blood from the left side of the heart is pumped through the minute vessels 
of the rest of the body before its return to the right side, where it is dispatched 
to the lungs. As a result of Galileo's experiments, the pump had become 
an object of scientific interest. Fabricius, under whom Harvey studied at 
Padua, had discovered that the veins have watchpocket valves which can 




FIG. 246. HARVEY'S FIGURE TO ILLUSTRATE His EXPERIMENT 
(From Singer.) 

only let blood pass one way, as the valves of a common pump only let water 
pass one way (Fig. 244). Fabricius was a colleague of Galileo. Harvey was 
a fellow-student with Torricelli. 

A conclusion which rests on so few and such simple experiments might 
have been made by the physicians of antiquity, if an understanding of the 
way in which a pump-valve acts had been part of their social culture. With 
the discovery of the continuous circulation of the blood a long familiar fact 
about the body was endowed with new interest. The "venous" blood which 
returns to the right side of the heart from, the tissues and passes thence to 
the lungs is dark purple. The "arterial" blood which issues from the lungs, 
and is pumped by the heart into the main arteries, is bright scarlet. The 
blood undergoes a definite and very obvious change through coming into 
contact with the air taken into the lungs. 

One of the first experiments which Boyle demonstrated to the Fellows of 



The Rebirth of Materialism 413 

the Royal Society was the fact that an animal dies at once when all the air 
is exhausted from a vessel in which it is placed. He also showed that a lighted 
candle is extinguished by the same treatment, and that sulphur and charcoal 
refuse to burn if heated in a vacuum, though they burst into flames when air 
is admitted. Gunpowder, which is a mixture of nitre (saltpetre), sulphur, and 
charcoal, did not behave in the same way. In a vacuum it was found to burn 
readily. Hooke proved that animals can be kept alive, when their normal 
respiratory movements are prevented, if air is blown through the lungs with 
bellows. He also showed that the respiration of an animal or the burning of 
a flame lasts longer in a large than in a small closed space containing air. 
Lower, another member of the same coterie of English men of science, 
showed that dark "venous" blood obtained by cutting a vein becomes bright 
scarlet when shaken with air. Thus the change of colour was due to "the 
particles of air insinuating themselves into the blood." In an early publication 
of the Royal Society, Slare (1693) develops the conclusion a little further by 
analogy with experiments in which solutions of copper compounds changed 
colour in a vacuum. Curiously enough he did not carry out the simple experi- 
ment to show that arterial blood acquires the purple hue of venous blood 
when shaken in an exhausted vessel connected with an air-pump. This was 
done by Priestley a century later. 

The power of gunpowder to ignite when heated in a vacuum suggested to 
Hooke that combustion depends on the combination of substances with a 
constituent common to air and nitre. Proof was provided by Mayow, a 
contemporary English physician (circa 1670). Mayow adapted an experiment 
which had been carried out centuries before his time. In classical antiquity 
it was known that a candle burning in a space closed by inverting a glass 
vessel over water makes the latter rise in the vessel (Fig. 246). Mayow repeated 
this experiment, substituting a mouse for a candle, with the same result, 
i.e. the water level rose. The mouse eventually died, just as the candle-flame 
is eventually extinguished. Thereafter the residual air will no longer support 
life or combustion. 

Adopting the atomistic views of Hooke, Mayow drew the radical conclusion 
that air consists of two sorts of particles. One sort is taken up by the lungs, 
or used by the candle-flame, and another is incapable of taking part in com- 
bustion or respiration. Having shown that the residual or foul air could be 
made suitable for breathing, or combustion by heating the nitre (or saltpetre) 
used in making gunpowder, he concluded rightly that respiration and com- 
bustion both result in removing the same "nitro-aerial particles." These 
nitro-aerial particles, or as we now say atoms of oxygen, make it possible for 
gunpowder to explode in a closed space. Thus air is not a simple substance, 
or element incapable of further dissolution. It contains at least two gaseous 
constituents, one being present in saltpetre. If Mayow's fundamental experi- 
ment had been carried out with an air-trap of mercury instead of water, no 
rise of level would have happened as the air lost its power to support life 
and flame. Mayow showed that air was deprived of one of its constituents 
without realizing that this constituent was replaced by another gas, the 
characteristics of which were first studied by Black (1754) in the middle of 
the eighteenth century. 



414 



Science for the Citizen 



Van Helmont was led to invent the word gas by discovering that the 
bubbles produced in fermentation and the "air" in which charcoal had been 
burned to ashes, are both unable to support combustion. Hence they are not 
identical with ordinary air. The "gas sylvestre" which van Helmont collected 
from the wine vat was found to turn a solution of lime milky. Black found that 
the bubbles of gas given off when acids are poured on chalk, and the air we 
breathe out, both have the same characteristic. He thus showed that the 
oxygen which is taken from the air by breathing or by the combustion of 
charcoal is replaced by another gas, which he called fixed air. This fixed 



FIG. 246. BURNING A CANDLE IN A CLOSED SPACE 

To measure the volume of air at the end of the experiment the glass jar must be 
depressed till the level of the water inside it and outside it is the same. The residual 
gas in the vessel is then at the same pressure (atmospheric) as at the beginning of the 
experiment. 

air, which we now call carbon dioxide or carbonic acid gas, is unlike the air 
we breathe into our lungs in many ways. For instance, it dissolves readily 
in water, with which it forms a slightly acid solution. Hence it disappeared in 
Mayow's experiment, leaving the air less dense. The water therefore rose to 
equalize the pressure. 

Black was a professor in Glasgow, and later in Edinburgh University. His 
many fundamental discoveries in chemistry and physics will be referred to 
again and again. He was the central figure of a brilliant coterie of Scotsmen 
about whom more will be said. The activities of this circle show how theo- 
retical science continually renews its youth by the infusion of new problems 
derived from the common experience of mankind and from contemporary 
social needs. The immediate problem which prompted Black's researches 
on lime is indicated in the biographical sketch accompanying Kay's Original 
Portraits: 



The Rebirth of Materialism 415 

We are informed by himself that he was led to the examination of the 
"absorbent earths partly by the hope of discovering a new sort of lime and lime- 
water which might possibly be a more powerful solvent of the stone than that 
commonly used. The attention of the public had been directed to this subject 
for some years. Sir Robert, as well as his brother Horace, afterwards Lord 
Walpole, were troubled with the stone. They imagined they had received 
benefit from a medicine invented by Mrs. Stephens, and, through their interest 
principally, she received five thousand pounds for revealing the secret. It was 
accordingly published in the London Gazette on June 19, 1739. This had 
directed the attention of medical men to the employment of lime-water in 
cure of the stone. 

The eminence of the Walpoles would not have ensured immortality to 
Black if other and more important circumstances had not drawn attention 
to his work. Shyness and ill-health prevented Black from publishing his 
important discoveries on heat. They were communicated verbally to the 
Newtonian Society of Edinburgh, and they would have passed unnoticed, if 
they had not inspired the invention of Watt's steam engine (see pp. 584-89). 
His friendship with Dr. Roebuck, who started large-scale commercial manu- 
facture of sulphuric acid, brought Black into contact with social forces which 
could find a use for new knowledge. But for that his discovery of fixed air 
would have been buried in the medical thesis in which it was announced. 

Dr. Roebuck, a physician from Birmingham, established a sulphuric acid 
factory at Prestonpans in 1749, started the famous ironworks at Carron in 
Stirlingshire in 1760, and leased the Duke of Hamilton's coalfield to use pit 
coal for reducing iron ores. Scotland, which had lagged behind England in 
the exploitation of its mineral resources, had begun to develop rapidly in 
this direction after the Act of Union and the Stuart rebellions. Coal mining 
was extended in Fifeshire and in the Lothians, where the Newcomen engine 
was introduced for pumping. In 1755, when an Edinburgh Society was 
established to promote the arts and manufactures, social conditions in 
Scotland were comparable to those in England when the Royal Society was 
founded. Industry was primarily concerned with the search for materials, 
and chemistry was therefore in demand. 

Mechanization had begun in the coalfields, where Newcomen's atmo- 
spheric engine was in use, when Black and Roebuck provided financial backing 
for a young technical assistant in Glasgow University to improve its design. 
In 1765 Watt is carrying out experiments on their behalf to test a process for 
making alkali by the decomposition of lime and sea salt. Hamilton (The 
Industrial Revolution in Scotland) tells us that the middle of the eighteenth 
century was also a period of rapid agrarian development in Scotland. Francis 
Home, Professor of Materia Medica and a prominent figure in the Edinburgh 
circle, is busy finding out "how far chymistry will go in settling the principles 
of agriculture." His treatise was published (1775) a year after the presentation 
of Black's thesis. Interest in capitalist farming had been an outstanding 
feature of the programme which the English Royal Society had undertaken 
in the first decade of its existence, when Glauber had lately shown that 
saltpetre is the "active principle" of manure. These early researches on soil 
chemistry led Francis Home to important discoveries which laid the found- 



416 



Science for the Citizen 



ations of commercial soil chemistry. A year after the publication of his 
Principles of Agriculture he received a gold medal from the "Honourable 
Board of Trustees for the Improvement of Manufactures in North Britain" 
in recognition of his "Experiments on Bleaching." We are told in Kay's 
Portraits that "he received many testimonies of eminent manufacturers whose 
art it had much improved." He was instrumental in introducing sulphuric 
acid for use in bleaching linen, and thus created a new commercial demand 
for the parent substance of modern chemical industry. 



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FIG. 247. MEASURING THE VOLUME OF A GAS 

In measuring the volume of a gas in a graduated glass vessel over a fluid, it is important 
to depress the latter till the level of the fluid inside and outside it is the same. If not, 
there will be a pressure difference between the gas and the outside air depending on 
the height to which the water rises. Suppose in the first cylinder (i), of sectional area 
a sq. cm., the length of the empty column is x cm., then its volume is ax c.c. If the 
oxygen is used up, e.g. by burning magnesium in it, the residual volume is \ax> and 
the height of the dead space $x when the pressure is the same as before (i.e. in equili- 
brium with the atmosphere). To make it equal (iii) the cylinder must be pushed down 
a little. Otherwise, the fluid will rise as it has risen y cm. in (ii). The volume of the dead 
space in (ii) is therefore a(x y}. If the fluid is mercury and the atmospheric pressure 
is P cm. of mercury, the difference in pressure between the outside air and air in (ii) 
is y cm u of mercury, and the pressure of air in (ii) is therefore P y cm. of mercury. So 

- k = (P - jvM* - y) 



The only unknown quantity here is y, if the barometric pressure is known, and the 
height to which the fluid will rise is calculable. If the fluid is water the numerical value 
of P is 13-6 times the figure for the mercury barometer. 



In Black's researches on lime the Aristotelian belief that chemical changes 
result from combinations of the "volatile" and imponderable elements air 
and fire with weighable matter in its elemental forms of water and earth , 
is making way for the modern view. Black found that fixed air is given off 
when chalk is heated to form quicklime, and estimated the quantity of fixed 
air which is combined with lime (calcium oxide) to form chalk by weighing 
the quicklime produced when a weighed quantity of chalk was heated. He 
determined how much chalk and how much quicklime are required to 



The Rebirth of Materialism 417 

neutralize an acid, i.e. to make the acid no longer able to change the colour 
of a dye like litmus, and found that the weight of lime which neutralizes an 
acid without producing effervescence is equivalent to the weight of lime 
produced by heating the amount of chalk required to neutralize the same 
amount of acid. Hence the gaseous constituent combined with Ume in chalk, 
and liberated as bubbles when acid is poured on the latter, does not affect 
its power to neutralize an acid. 

The interest which Black's researches aroused in men of Roebuck's type 
was partly due to the fact that commercial production of alkali was becoming 
an imperative need of textile industry. The older chemists had used the word 
alkali for any substance which neutralizes an acid. Quicklime, potashes, and 
caustic potash, obtained by boiling a solution of potashes with lime, were all 
"alkalis." Black showed that the potashes of industry (potassium carbonate), 
like chalk, liberate fixed air when acted on by acids. His experiments proved 
that the way in which chalk and potashes are built up has more in common 
than the way in which quicklime and potashes are built up. We express this 
today when we say that both belong to the class of compounds called car- 
bonates. 

Two new methods of enquiry were emerging. Each had tremendous 
consequences for the future of man's command over the materials at his 
disposal. Hitherto chemists had classified the qualities of substances by their 
uses, and had neglected the observed quantities in which they combine. About 
this time they begin to classify them by their constituents, and to study the 
proportions in which their constituents are combined by weight. The former 
makes it possible to make a comprehensive survey of the sources of materials 
for social use. The second tells us whether the yield we can expect from a 
particular source justifies the effort expended. One difference between the 
chemistry of today and the chemical art of the Middle Ages lies in the fact 
that it can tell us all the possible sources from which we can get material 
substitutes and the yield we can get by using them. 

Medical men and pharmacists like Black, Francis Home, and Scheele were 
prominent among the theoretical leaders who prepared the way for the rise 
of chemical industry, and Roebuck and Keir, two of the leading entrepreneurs 
of the period (vide infra), were medically qualified. Since chemistry was not 
as yet separated from the practice of medicine it inevitably benefited from 
the contemporary revolution in biological classification. The classification 
of plants arose out of the social practice of ancient medicine. Commer- 
cial horticulture, seed production for agriculture, and the systematic policy 
of surveying the unexploited wealth of new countries during the period 
of colonial expansion which intervened between the discovery of the New 
World and Cook's Australian voyage, conspired to give a new and powerful 
impetus to biological classification. This reached its zenith in the Systema 
Naturae of Linnaeus published in the middle of the eighteenth century. 
Linnaeus set forth a classification of the "mineral kingdom" along with that 
of animals and plants. In this he followed a practice which grew out of the 
instructions which Elizabeth issued to her sea captains. The proceedings of 
learned academies from 1650 to 1670 contain innumerable miscellanies of 
local information about plants, animals, or minerals, communicated from 

o 



418 Science for the Citizen 

colonies, like Ceylon or Malabar. Such accounts were often based on collec- 
tions made with direct encouragement from colonial governors. Although 
Linnaeus himself did not embody the results of new knowledge about 
chemical composition in his classification of the mineral kingdom, it was 
important because it focussed attention on a new theoretical need. 



EXPLOSIONS 

Let us now see how researches suggested by the properties of gun- 
powder and by explosions in coal-mines conspired to stimulate other dis- 
coveries about the individuality of gases. The former encouraged the study 
of the gas called sulphur dioxide. Sulphur and charcoal both burn in air. If 
they are impure there is a small solid residue of salts. If sufficiently pure they 
leave no solid residue. The burning of sulphur is accompanied by the form- 
ation of very pungent fumes, which dissolve in water to form a solution 
which affects vegetable dyes, such as litmus, in the same way as the mineral 
acids. Similar fumes are produced in the absence of air when sulphur is 
heated with another constituent of gunpowder saltpetre or nitre. The 
explanation of this lies in another fact. When heated alone, nitre gives off 
a "gas" oxygen which sustains combustion more readily than air itself, 
and makes foul air shaken with water suitable for breathing. When nitre is 
mixed with sulphur and charcoal, this gas is taken up by the sulphur and 
charcoal with the formation of two gases, fixed air (carbon dioxide), and 
fumes of burnt sulphur (sulphur dioxide). An explosion is nothing more 
than the sudden change of volume which occurs when these gases are liberated 
in a closed space. Put in modern phraseology, these were essentially the 
conclusions to which Boyle's experiments with his air pump led him. In his 
famous book, the Sceptical Chymist, he roundly attacked the doctrine accepted 
by nearly all his predecessors who interpreted the use of heat to facilitate 
chemical changes as proof that fire dissolves complex substances into simpler 
ones. He rejected the belief that there were only a few elementary substances 
which participate in chemical processes. 

A clue to the nature of explosions in coal-mines was found when Clayton 
(1691) prepared an "inflammable air" by heating coal in a retort. A few years 
later the physician Stephen Hales, who first made experiments on the pres- 
sure of the blood, recorded the observation that 158 grains of Newcastle 
coal yield 180 cub. in. of the new gas. Coal gas is not a single chemical sub- 
stance of constant composition. Though its discovery did not, therefore, 
intrinsically add to the existing stock of knowledge about how substances 
combine, it presented two arresting features which quickened the growing 
recognition that a great variety of different substances exist in the gaseous 
state. It was inflammable and lighter than air. The search for other gases 
received a powerful impetus from its novelty, and was also reinforced by the 
fact that acids were now being manufactured on a commercial scale. In the 
middle of the eighteenth century Cavendish made a thorough examination 
of the bubbles given off from the action of strong acids on metals. He intro- 
duced a simple device (Fig. 248) for collecting them, and thus discovered a 
new inflammable gas which proved to be much lighter than air or coal gas. 



The Rebirth of Materialism 419 

Coal gas burns in a closed space with a sooty flame. In abundant air, water 
vapour is produced, and the residual gas turns lime water milky. The gas 
that we now call hydrogen, which Cavendish (1766) obtained from the action 
of sulphuric acid on zinc or of hydrochloric acid on tin, burns without the 
production of soot or fixed air (carbon dioxide). The only product of its 
combustion is steam, i.e. water in the gaseous state. 

The discovery that water is a compound substance was made by Priestley 
two years after he joined the circle of James Watt the inventor, in 
Birmingham (see page 431). The Aristotelian catalogue of elements, from 
which air had been removed, therefore sustained a second rude shock 
from the discovery of hydrogen. Elemental water could now be made by 




FIG. 248 

Apparatus for collecting hydrogen gas by the action of hot, strong hydrochloric acid 
(introduced through the thistle funnel) on tin, or of sulphuric acid on zinc. The 
cylinder in which the gas is collected over water is at first completely full of water. 
To collect a water-soluble gas like carbon dioxide or ammonia, mercury or paraffin 
should be used instead. 

combining hydrogen, a constituent of acids, with oxygen, a constituent of 
elemental air. 

The temper of the times had moved far from witch burning when a new 
secular miracle added to the prestige of chemical science. Black had demon- 
strated the ascent of toy balloons filled with coal gas, and subsequent events 
soon brought its novelty into the arena of everyday life. In the year 1782 
two Frenchmen, the brothers Montgolfier, devised a startling demonstration 
that hot air is less dense than cold. A large silk bag with an opening at the 
bottom was held by ropes over a bonfire. When it was deemed to be suffici- 
ently full of hot air, the ropes were released, and the first balloon made its 
ascent. A year later a similar attempt was made, this time with a carrier in 
which a duck, a hen, and a sheep were sent up. Owing to the carelessness 
of the sheep, the results were fatal to the hen. Otherwise the experiment was 
a triumph. In the same year Rozier made an ascent, traversing a distance of 



420 Science for the Citizen 

over five miles, in a hot-air balloon. Meanwhile Charles had constructed a 
large hydrogen balloon in Paris. A great concourse collected at the Champs- 
Elysees to witness the spectacle. His success made it possible to remain in the 
air without fear of being forced to descend owing to cooling. In 1784 
crowds collected in the Strand to see the first ascent in a hydrogen balloon 
by Lunardi, and a year later Blanchard crossed the channel. The Soho 
group, to which we shall refer later on, took an active interest in the possi- 
bilities of the invention. Prosser tells us that John Southern, a Fellow of the 
Royal Society, employed by Boulton and Watt, 

was the author of a tract on balloons, published at Birmingham in 1783 at 
the time when the ascents by the French aeronauts were exciting much 
attention. 

HEAT AND WEIGHT 

The belief that earth is an element was never more than a figure of speech. 
It had made way for a more literal statement of fact long before air had been 
recognized as a complex substance with weighable constituents, of which 
one is also present in water. The elemental nature of fire was the last of 
Aristotle's catalogue to succumb. The materialism of Democritus had drawn 
a clear and correct distinction between heat which accompanies motion 
or, as we should say, friction and heat which accompanies chemical changes, 
which may also be associated with light when we see a flame. According to 
the materialistic doctrine of the Greek atomists, heat, like light, is merely 
one of the ways in which our sense organs detect the presence of matter. 
Such a view was far too sophisticated for the beliefs of a primitive civilization. 
Fire had a host of hallowed associations calendrical, sacrificial, and sexual 
in all the ancient mystery religions. Children were passed through the 
fire to Moloch, and virgins attended the sacred flame which never failed. The 
divine fire was also the logos spermatikos, the light that Ughteth every man. 
In Platonism, Stoicism, Gnosticism, and Mithraism, from which Christian 
theological ritual severally derived so many of its ingredients, fire was an 
object of veneration, and a symbol of unspeakable mysteries. It is little matter 
for surprise that Fire was the last of Aristotle's elements to go. 

It retained its hold most tenaciously in the oldest branch of chemical 
technology. Two basic processes of metallurgy had persisted unchanged 
from the dawn of the Iron Age till the introduction of pitcoal to replace 
charcoal. For extraction of some metals, it was sufficient to roast the ore or 
calx with charcoal in a closed space. Chalk was also added as a "flux" to 
combine the quartz and clay present in the ore into a fusible glass which 
could be easily separated from the metallic mass. Another source of metal, 
the sulphureous ores, could only be extracted by previous roasting in a current 
of air. This was accompanied by the evolution of sulphureous fumes (sulphur 
dioxide). On subsequent reduction with charcoal the calx formed from the 
sulphureous ore yielded the metal. Alchemical experiments with metals 
had added three important items of additional information: 

(a) Some metals, especially tin and mercury, could be converted into 
calces (oxides) by heating strongly in air. 



The Rebirth of Materialism 421 

(i) The formation of the calx is accompanied by increase of weight. 

(c) Some metals, e.g. iron, yield sulphureous ores (sulphides) when strongly 
heated with sulphur in a closed space. 

In addition to the seven metals associated with the seven pre-Copernican 
"planets" two others, antimony and zinc, had been worked as early as 1000 B.C. 
Thereafter practically no progress was made in practical metallurgy, and 
no new facts of theoretical importance, other than the three stated above, had 
been ascertained till towards the end of the eighteenth century. In ancient 
metallurgy the use of charcoal to reduce the oxide ores, and the preliminary 
treatment of sulphureous ores in the blast furnace, were empirical facts. 
Their significance resided in the mysterious properties of fire. Failure to 
form correct conclusions about the way ores are built up made it impossible 
to lay down rules for seeking new sources and sorts of metal. 

In the light of what we have now learned about combustion and explosion, 
it is easy to see what happens in the extraction of metals. The sulphureous 
ore is a compound (sulphide) of metal and sulphur. The sulphureous fumes 
(sulphur dioxide) and calx (metallic oxide) formed in the roasting furnace are 
compounds of oxygen with sulphur and metal respectively. The air of the 
blast furnace supplies the oxygen (or nitre-aerial particles) which converts 
the sulphur of the ore into sulphur dioxide, and the metal of the ore into 
metallic oxide. The charcoal or coke used in reducing the calx is consumed 
when it burns, forming fixed air (carbon dioxide) by combining with oxygen. 
In a closed space it cannot take this oxygen quickly from the air. It takes it 
slowly from the ore, as it takes it rapidly from the nitre when gunpowder 
ignites in a vacuum. The formation of a calx from a metal when heated in 
air is essentially like the combustion of charcoal to form fixed air. The only 
difference is that charcoal ignites at comparatively low temperatures, whereas, 
with a few exceptions like the magnesium of flash-light photography, metals 
only combine rapidly with oxygen at very high temperatures. The entire 
sequence of changes when metals are extracted will be made clear if we now 
separate the known facts from what we now know to be the correct explan- 
ation. 

The known facts were: 

(1) sulphur (solid) + metal (solid) =- sulphureous ore (solid) 

(e.g. iron) (e.g. pyrites) 

(2) sulphur (solid) + oxygen (gas) = sulphureous fumes (gas) 

(of the air) (sulphur dioxide) 

(3) metal (solid) + oxygen (gas) = calx (solid) 

(of the air) 

(4) charcoal (solid) + oxygen (gas) = fixed air (gas) 

(of the air) (carbon dioxide) 

The likely explanation of the industrial processes was therefore 

(5) In the roasting furnace, 

sulphureous ore + oxygen = calx + sulphur dioxide. 

(6) In the closed furnace, 

calx + charcoal = metal + carbon dioxide. 



422 Science for the Citizen 

For complete proof of this it was necessary to show: 

(a) that the increase of weight when the calx is formed from its metal is 
due to the disappearance of an equivalent weight of oxygen; 

(b) that the gases formed in the roasting and closed furnaces were respec- 
tively identical with those formed when sulphur and charcoal burn in 
air; 

(c) that the difference between the weight of calx used and metal formed 
is equal to the difference between the weight of fixed air produced 
and the weight of carbon used up. 

The first step (a) had already been surmised by a French chemist Rey (1630), 
who had absorbed Galileo's doctrine,, before Boyle, "with the help of our 
engine/' had shown how he could "weigh the aire as we weigh other bodies 
in its natural or ordinary consistence." "Let all the greatest minds in the 
world," said Rey, "be fused into one mind, and let him seek diligently on 
the earth and in the heavens; let him search into every cranny of nature: 
he will only find the cause of this augmentation in the air." The last step (c) 
was made possible by Black's discovery. Carbon dioxide turns a solution 
of lime milky because it precipitates the insoluble compound calcium car- 
bonate (chalk). Black had found the weight of carbon dioxide in a given 
quantity of chalk. So the quantity of carbon dioxide in a given quantity 
of air can be ascertained by shaking it with lime water, filtering off the 
precipitate, drying the latter and weighing it. 

Even the English physicists, whose "nitro-aerial particles" offered the 
necessary clue to the nature of combustion, were readier to recognize its 
kinship to the breath of life than to probe into the Promethean secrets of 
metallurgy. Boyle himself stuck to the belief that metals like tin and lead, 
which readily form oxides when heated in air, increase in weight by absorbing 
the fiery particles with power to penetrate the walls of the furnace. Mean- 
while a school of continental chemists preserved the purity of their studies 
from contamination with the growing knowledge of the nature of heat by 
fabricating a doctrine which may well commend itself to those economists 
who believe in the possibility of erecting science on a foundation of self- 
evident principles. The doctrine of phlogiston, which was the last attempt 
to sustain the elemental nature of fire, was concocted towards the end 
of the seventeenth century. It provides an instructive example of the way 
in which facts may be used to illustrate instead of to test the truth of a 
theory. The argument runs as follows. It is self-evident that if things burn, 
they must contain the fire principle. A combustible substance is, therefore, a 
combination of a calx or non-combustible material with the fire principle 
phlogiston. The escape of phlogiston when a combustible substance burns is 
accompanied by production of incombustible material which actually weighs 
more than its predecessor. It is therefore self-evident that phlogiston must 
be endowed with the opposite of weight, levity, or the power to make a body 
weigh less. Much valuable time was wasted in disproving a theory with 
nothing to commend it but the elegance of flawless reasoning from premises 
which have no foundation in fact. 
What contributed most to discredit phlogiston in the long run was 



The Rebirth of Materialism 423 

growing interest in the nature of heat. Heat, as a source of mechanical 
power, was beginning to effect a veritable revolution in human life. The 
problem of measuring heat and of classifying the different sources and means 
of transmitting it was beginning to eclipse the physical problems which had 
arisen from the practice of time reckoning and of earth survey. As we all 
know, the production of heat alone or of heat and light together can be 
brought about without any other change in the properties of a body. A 
poker which is heated till it glows is in other respects just a piece of iron 
weighing as much as it did before. The heat which is accompanied by incan- 




A< I Air burning 
ift'CoaLiS' 



Coalgas 



\ air 



FIG. 249. APPARATUS SHOWING Am BURNING IN COAL GAS 

descence when a body is said to burn away only differs in intensity from 
the heat given out or taken in in any change from one substance to another. 
It is the same physical phenomenon as the heat produced by friction when 
a wheel is not lubricated, or the heat acquired by contact when the poker is 
put in the fire. 

All chemical changes are associated with difference of temperature. 
When washing soda (sodium carbonate) is dissolved in water the solution 
is cooler than the surrounding air. When sulphuric acid is added to water 
the mixture is hotter. If we let sulphuric acid trickle through a narrow 
tube immersed in a large bath of water, the fluid round the orifice will get 
hot. If we let water trickle through a narrow tube into a bath of sulphuric 
acid, the fluid near the orifice becomes exceedingly hot. Similarly air will 
burn in coal gas just as coal gas will burn in air (see Fig. 249). When we 
speak of a combustible substance, or a substance which supports combustion, 
we are using relative terms, assuming that we are carrying out an experiment 
in particular conditions, 



424 



Science for the Citizen 



CHARLES' LAW 

The hot air which makes the Montgolfier balloon go up is not less dense 
because it absorbs "fiery" particles endowed with the peculiar gift of levity 
from the matter burned to inflate it. Hot air is less dense, because the 
same number of particles occupy more space. The volume occupied by a 
given weight of a gas whose other characteristics remain unchanged increases 




FIG. 250 

Apparatus for finding effect of temperature on the volume of a gas at constant pressure. 
The tube A can be raised and lowered so that the level of mercury in the tubes A and B 
is always the same. Thus the volume of gas, shown by the scale of the graduated limb 
B, is always at atmospheric pressure when the measurements are made. The tempera- 
ture is changed by putting warm water into the glass jacket C. 

according to a definite rule as the temperature is raised. This rule was first 
discovered by the inventor of the hydrogen balloon. 

Boyle's Law is not a satisfactory guide for calculating the weight of a gas 
from its volume, unless the temperature remains the same. So in Boyle's 
time accurate weighing was hampered because a satisfactory thermometric 
scale had not been fixed (see Chapter XI). The rule, which is called Charles' 
Law, provides us with an opportunity of illustrating how graphical methods 
are used in science to detect the rule which connects a set of observations. 
The changes in volume which a fixed weight of gas undergoes when the 



The Rebirth of Materialism 425 

temperature is raised and the same pressure is maintained, can be recorded 
by the simple apparatus shown in Fig. 250. The results of such an experi- 
ment are plotted in the next figure. The distance of each point measured 
along the #-axis corresponds to the volume of gas at a particular pressure. 
The latter is represented by a distance measured along the y axis. Thus 
the line BC represents the difference between the volume of gas v measured 
at f on the centigrade scale and the volume V (= AO) of the same sample 
of gas measured at C., i.e. BC = v V. The line AB is the corresponding 
temperature difference t = t. 

Since all the observations fall approximately on a straight line, the ratio 
BC ~ AB or (v V) t is approximately the same for corresponding 
values of v and t included in the observations made on the same sample 
at the same pressure. Since V is fixed in any such experiment the ratio 
[(v V) -r t] -T- V is also fixed in any particular experiment. This ratio 
is found to have the same numerical value, approximately -^rs 9 in different 
experiments with different cases, i.e. 




V 

V "~ l ^ 273 



* 273+ t 
1 + 273 = 273 

273- 



If we put T instead of (t + 273), the volume v of a gas at any temperature 
t is directly proportional to T. 



The ratio V 4- 273 involves two fixed numbers, since V is the volume of 
gas at C. in any particular experiment. Throughout a single experiment 
it may thus be replaced by one constant thus 

v= CT 

The number T obtained by adding 273 to the temperature registered by a 
centigrade thermometer is called the absolute temperature of the gas. At 
constant pressure, the ratio of the volume of a given weight of gas to its 
absolute temperature is fixed, just as at a fixed temperature the product of 
the pressure and volume of a given weight of gas is constant. The volume 
of a gas at constant pressure is increased by 2^3 of its volume at C. for 
each degree rise in absolute temperature. 

By applying Boyle's Law and that of Charles, the weight of gases partici- 
pating in chemical processes can be measured with the greatest ease. Their 



426 



Science for the Citizen 



densities at a standard temperature and pressure must first be tabulated once 
and for all for reference. The densities of gases determined directly by 
weighing an evacuated vessel, filling it with gas at standard pressure and 
reweighing it, are usually recorded for the standard temperature C. 
(273 on the absolute scale), and a standard pressure of 760 mm. on the 
mercury barometer (called I atmosphere). The weight of any gas can then be 
calculated from its volume recorded in a graduated glass vessel (Fig. 247) 
at a known temperature and atmospheric pressure. The following problem 
illustrates how the gas laws are applied to save the trouble of weighing a gas. 
Suppose the chemist who wants to know the proportions in which magne- 




2-0 



1-5 



O-5 




-2O 



-10 



+10 



+20 (2/-3) 30 +-40 



FIG. 251 

Graphical representation of change of volume which a quantity of gas occupying 
1 cub. ft. at C. undergoes as the temperature is raised, the pressure at which the 
gas is measured being constant (atmospheric) throughout. 

sium combines with oxygen to form its "calx," has found that 1 123 grams 
of magnesium combine with 560 c.c. of oxygen measured in a graduated vessel 
when the mercury barometer reads 75 cm. and the temperature is 15 C. His 
tables of density tell him that 1 litre of oxygen weighs 1 -43 grams at S.T.P. 
(0 and 760 mm.). By applying Boyle's law (page 387), which tells us that 
volume is inversely proportional to pressure, he can find how much space 
the gas would occupy at standard pressure. This is 



560 x 75 
76 



= 552 -6 c.c. 



To apply Charles' law he converts the centigrade temperature 15 C. to the 
absolute scale, 273+ 15= 288. Since C. on the absolute scale is 273, 
the volume of gas measured at standard temperature is 

552-6 X in- 523 -8 c.c. 



The Rebirth of Materialism 427 

Hence the measured volume of 560 c.c. at 15 C. and 750 mm. would occupy 
523-8 c.c. at S.T.P., and therefore weighed (523-8x1-43)- 1,000=0 -749 
grams. Thus 0-749 grams of oxygen combine with 1-123 grams of magne- 
sium, and the combining ratio is approximately 2 : 3. 

Graphs which are superficially like the one shown in Fig. 251 are exhibited 
in textbooks of economics to illustrate theories about supply and demand, 
or wages and profits. This makes readers who lack self-confidence infer that 
economics is an "exact" science. Two characteristics of the graph in Fig. 251 
should therefore be recognized. One is that it tells you how to do something , 
i.e. to find the weight of a given volume of gas by reference to tables of density. 
The other is that each point on it corresponds to two actual measurements. 
The draughtsmanship exhibited in textbooks to illustrate marginal utility 
has neither of these characteristics. Neither of his co-ordinates corresponds 
to a measured or, in the present state of knowledge measurable entity 
when Dr. Hicks (Theory of Wages) states: 

If now the employer's concession curve cuts the resistance curve on the 
horizontal part, the Union will generally succeed in maintaining its claim; but 
if it cuts it at a lower point, compromise will be necessary and it is over such 
compromises that misunderstandings and strikes most easily arise. 

THE CALCINATION OF METALS 

The recognition that the roasting of a sulphureous ore in the furnace, the 
formation of a calx or oxide when a metal is heated in air, or the slower rusting 
of iron in moist air, each involve processes akin to the explosion of gun- 
powder, to the burning of charcoal, or to the respiration of a mouse, followed 
quickly after the work of Black and Cavendish. Shortly after the discovery 
of hydrogen, a clearer insight into the process of oxidation, that is to say, 
chemical change in which oxygen enters into a new combination with an 
element, was gained through the complete separation of both the principle 
constituents of air. Rutherford (1772) used up all the oxygen (roughly one- 
fifth of the total volume) by burning in air substances like charcoal, sulphur, 
and phosphorus, recently prepared by the Swedish chemist Scheele from 
bone ash. When this is done, the acid products (oxides of carbon, sulphur, 
and phosphorus), being soluble, are easily removed by shaking the residual 
gas with a mildly alkaline solution. What remains forms four-fifths of the air 
by volume. It can be most readily prepared by burning magnesium in air 
till no more magnesium will burn. The oxide of magnesium, being a solid 
powder, settles on the sides of the vessel, leaving a residual gas which is 99 
per cent nitrogen. Nitrogen neither burns like hydrogen, nor allows sub- 
stances to burn in it as oxygen does. Like oxygen and hydrogen it is colourless, 
odourless, and only sparsely soluble in water. It is but little less dense than 
air itself. 

The separation of this relatively inert portion of the atmosphere was 
followed by a careful study of the characteristics of its active partner. As 
early as 1489 the alchemist de Sultzbach had noticed a "spirit" given off 
when red oxide of mercury is heated, leaving the metal behind. Shortly after 
Hooke's work several chemists had collected "fire air" liberated by heating 
saltpetre. Apart from noting that combustible substances burn more fiercely 



428 Science for the Citizen 

and brightly in it, none of these pioneer claimants to the "discovery" of 
oxygen made a careful study of the characteristic properties which distinguish 
the gas from the rest of the atmosphere. Simultaneously in the year after 
the separation of nitrogen, three chemists, Scheele, a Swedish apothecary, 
Priestley in England, and Lavoisier in France, showed the identical character- 
istics of a gas produced from several different sources. They also established 
its identity with the "nitro-aerial particles" of common air by recognizing 
the same products of combustion, such as (a) carbon dioxide which turns 
lime water milky, (6) sulphur dioxide and phosphorus pentoxide, each with 
its characteristic odour, and (c) the solid reducible metallic calces like iron rust. 

Although the elementary nature of air had been tacitly abandoned by 
Boyle and Hooke, chemists had found it hard to realize how many kinds of 
matter exist in the gaseous state. For long it was the fashion to speak of the 
new gases which had been successively discovered as different kinds of air. 
The elementary gas hydrogen, having great levity and inflammability,* was 
inflammable air. The compound gas carbon dioxide formed by burning 
charcoal, heating chalk, or pouring acids on potashes, was fixed air. The 
pungent, highly soluble, and powerfully acid "spirits of salt" exhaled by 
distilling vitriol (sulphuric acid) with sea salt (sodium chloride) to make 
Glauber's salts was not yet called by its modern name. What we now call 
hydrochloric acid gas or hydrogen chloride was "marine acid air." The 
pungent highly soluble and strongly alkaline spirit prepared by heating lime 
with smelling-salts (ammonium carbonate) or sal ammoniac (ammonium 
chloride) and collecting the gas over mercury was "alkaline air." The 
resolution of the air into two distinct constituents, each with its own 
characteristic properties, finally put a stop to this confusion. Henceforth each 
gas had a name of its own. 

Every obstacle to an intelligible account of the metallurgical processes was 
now removed. After a century of futile sophistication, deductive chemistry 
succumbed to Lavoisier's experiments on calcination, and the phlogiston 
theory was abandoned. Lavoisier's experiments on the formation of the calx 
when tin is heated in air showed three things : 

(a) A fixed quantity of tin could be converted into calx by a fixed quantity 
of air, yet some air remained however much tin was used. The calx 
was, therefore, a combination of metal and one of the constituents of 
air in definite proportions. 

(b) A sealed retort containing the metal from which no air was allowed to 
escape weighed exactly the same before or after conversion of the 
metal and its calx. Therefore, the greater weight of the calx was not due 
to the penetration of "fiery particles" through the walls of the vessel. 

(c) The increased weight of the calx was exactly offset by the diminished 
weight of the air. The diminished weight of the air is not due to an 
increase of "levity." At fixed pressure it is accompanied by an actual 
decrease of volume. This was shown by the fact that air rushed in 
when the sealed retort was opened. The inrush of air was accompanied 
by an increase of weight, equivalent to the increase of weight when 
the same quantity of metal was converted into calx in an open vessel. 



The Rebirth of Materialism 429 

Similar results were obtained by heating mercury to its boiling point, 
when it forms the red oxide which can easily be converted back to its con- 
stituents as a source of oxygen. These experiments combined the essential 
features of Mayow's experiments with the mouse and the candle, with the 
additional information gained from weighing all the ingredients and products. 

THE RISE OF CHEMICAL INDUSTRY 

The death-blow to phlogiston was the need for precise guidance to meet 
the new social needs of chemical manufacture. The work of Priestley, Scheele, 
and Lavoisier, was undertaken when chemical manufacture in the modern 
sense was just beginning. In France, Lavoisier was one of a group of chemists 
which included Berthollet, who discovered the bleaching power of chlorine, 
and Leblanc, whose alkali process solved an acute technological problem 
of the revolutionary wars during the last decade of the nineteenth century. 
The encouragement which Lavoisier's work received from a growing 
demand for chemical knowledge in his time is sufficiently illustrated by the 
fact that he was appointed to direct the manufacture of gunpowder by Turgot 
two years after (1776) the completion of his researches on metallic oxides. 
He succeeded in making the saltpetre output of France fivefold greater, 
abolished the irksome regulations for collecting refuse and manure from 
private cellars, etc. (see page 407), and increased the explosive power of the 
mixture. He was later (1791) commissioned by the National Assembly to 
draw up a conspectus of the mineralogical resources of France. 

Priestley's parallel enquiries into metallic oxides are intimately related to 
the metallurgical problems of the Industrial Revolution which began in 
Birmingham. In 1760 Matthew Boulton set up a hardware factory employing 
over six hundred skilled workmen. The machinery was run by a waterwheel, 
for which the supply of water was insufficient in dry summers. Boulton 
conceived the plan of using a pump to return water from the outflow to the 
conduit. He had a keen interest in the nascent physical science of his time, 
fostered to some extent by a Scots physician named Small, to whom he had 
been introduced by Benjamin Franklin. Small was responsible for bringing 
Boulton into touch with James Watt, the young Scots engineer who had been 
working to improve the Newcomen design. From the partnership between 
Boulton and Watt in 1775 the new era of machine manufacture came into 
being. Boulton was a dose friend of Roebuck, whose first attempt to manu- 
facture sulphuric acid on a commercial scale had been made during his 
residence in Birmingham three years before he set up his factory with 
Garbett in Prestonpans. His keen interest in the chemical problems of metal- 
lurgy arose partly from the use of alloys in his factories. He was also a con- 
tractor for the Royal Mint. For a short time he was in partnership with Keir, 
who afterwards set up an alkali factory in Britain. 

The device which led to the partnership of Boulton and Watt may have 
been directly inspired by the propinquity of the Birmingham factory to 
the Potteries which provided an early market for the Boulton-Watt products. 
At that time the Potteries were importing their best clays from Cornwall. 
Between 1750 and 1760 two master potters of Staffordshire, John Turner 
and Josiah Spode, had also employed a Cornish steam pump to work a 



430 Science for the Citizen 

waterwheel in their industry, which relied for fuel on the North Staffordshire 
coalfields. Like Boulton, Wedgwood, the leading figure in the expansion of 
the Potteries, was an intrepid experimentalist. Like Boulton, he maintained 
a correspondence with Priestley, encouraged him to come to Birmingham, 
supplied him with free apparatus, and, in particular, took great pains to 
construct for him the best retorts and stoves. Wedgwood was elected a Fellow 
of the Royal Society in the same year as Priestley. After Priestley moved to 
Birmingham, Wedgwood co-operated with Boulton in providing financial 
resources to support him in his researches. 

The ensuing passage from Smiles^ biography of Boulton and Watt gives 
us a vivid picture of the cultural renaissance which accompanied their 
partnership in England : 

Towards the close of last century, there were many little clubs or coteries 
of scientific and literary men established in the provinces, the like of which 
do not now exist. ... At Liverpool, Roscoe and Currie were the centres of 
some such group; at Warrington, Aikin, Enfield, and Priestley of another; at 
Bristol, Dr. Beddoes and Humphry Davy of a third; and at Norwich, the 
Taylors and Martmeaus of a fourth. But perhaps the most distinguished of 
these provincial societies was that at Birmingham, of which Boulton and Watt 
were among the most prominent members. . . . The meetings were appointed 
to be held monthly at the full of the moon, to enable distant members to drive 
home by moonlight; and this was the more necessary as some of them such 
as Darwin and Wedgwood lived at a considerable distance from Birming- 
ham. . . . Dr. Darwin was regarded as the patriarch of the Society. His fame 
as a doctor, philosopher, and poet was great throughout the Midland Counties. 
He was extremely speculative in all directions, even in such matters as driving 
wheel-carriages by steam. . . .Dr. Priestley, the discoverer of oxygen and 
other gases, was one of the youngest. We find Boulton corresponding with him 
in 1775, principally on chemical subjects, and supplying him with fluor spar 
for purposes of experiment. Five years later, in 1780, he was appointed minister 
of the Presbyterian Congregation assembling in the New Meeting-house, 
Birmingham; and from that time forward he was one of the most active 
members of the Lunar Society. ... At the time when he settled at Bir- 
mingham, Priestley was actively engaged in prosecuting inquiries into the 
constitution of bodies. He had been occupied for several years before in 
making investigations as to the gases. The discovery of carbonic acid gas by 
Dr. Black of Edinburgh had attracted his attention; and, living conveniently 
near to a brewery at Leeds, where he then was, he proceeded to make experi- 
ments on the fixed air or carbonic acid gas evolved during fermentation. From 
these he went on to other experiments, making use of the rudest apparatus 
phials, tobacco pipes, kitchen utensils, a few glass tubes, and an old gun- 
barrel. The pursuit was a source of constant pleasure to him. . . . Such was 
Priestley, and such were his pursuits, when he settled at Birmingham in 1780. 
There can be little doubt that his enthusiasm as an experimenter in chemistry 
exercised a powerful influence on the minds of both Boulton and Watt, who, 
though both full of work, anxiety and financial troubles, were nevertheless 
found taking an active interest from this time forward in the progress of 
chemical science. Chemistry became the chief subject of discussion at the 
meetings of the Lunar Society, and chemical experiments the principal recrea- 
tion of their leisure hours. "I dined yesterday at the Lunar Society (Keir's 
house)," wrote Boulton to Watt; "there was Blair, Priestley, Withering, Galton, 



The Rebirth of Materialism 431 

and an American 'rebel/ Mr. Collins. Nothing new except that some of my white 
Spathos Iron ore was found to contain more air than any ore Priestley had 
ever tried, and, what is singular, it contains no common air, but is part fixable 
and part inflammable." To Henderson, in Cornwall, Boulton wrote, two months 
later, "Chemistry has for some time been my hobby-horse, but I am pre- 
vented from riding it by cursed business, except now and then of a Sunday. 
However, I have made great progress since I saw you, and am almost an adept 
in metallurgical moist chemistry. I have got all that part of Bergmann's last 
volume translated, and have learnt from it many new facts. I have annihilated 
Wm. Murdock's bedchamber, having taken away the floor, and made the 
chicken kitchen into one high room covered over with shelves, and these I 
have filled with chemical apparatus. I have likewise set up a Priestleyan water- 
tub, and likewise a mercurial tub for experiments on gases, vapours, etc., and 
next year I shall annex to these a laboratory with furnaces of all sorts, and all 
other utensils for dry chemistry." The "Priestleyan water-tub" and "mercurial 
tub," here alluded to, were invented by Priestley in the course of his investiga- 
tions for the purpose of collecting and handling gases; and the pneumatic 
trough, with glass retorts and receivers, shortly became part of the furniture 
of every chemical laboratory. 

Another passage from Smiles refers to the tangible support which Priestley 
received from Boulton and Wedgwood : 

Josiah Wedgwood was another member of the Lunar Society, who was 
infected by Dr. Priestley's enthusiasm for chemistry; and knowing that the 
Doctor's income from his congregation was small, he and Boulton took private 
counsel together as to the best means of providing him with funds, so as to 
place him in a position of comparative ease, and enable him freely to pursue 
his investigations. . . . Wedgwood had undertaken to sound Dr. Priestley, 
and he thus communicated the result to Boulton : "The Doctor says he never 
did intend or think of making any pecuniary advantage from any of his experi- 
ments, but gave them to the public with their results, just as they happened, and 
so he should continue to do, without ever attempting to make any private 
emolument from them to himself. I mentioned this business to our good 
friend, Dr. Darwin, who agrees with us in sentiment, that it would be a pity 
that Dr. Priestley should have any cares or cramps to interrupt him in the fine 
vein of experiments he is in the midst of, and is willing to devote his time to 
the pursuit of, for the public good. . . . Dr. Darwin will be very cautious 
whom he mentions this affair to, for reasons of delicacy which will have equal 
weight with us all. I mentioned your generous intention to Dr. P., and that we 
thought of 20 each; but that, you will perceive, cannot be, and the Doctor 
says much less will suffice, as he can go on very well with 100 per annum. 

The Darwin referred to in this passage was Erasmus, the grandfather of 
Charles, and author of the first book (Zoonomica) setting forth the evolutionary 
doctrine in Britain. The description of the Lunar Society given by Smiles 
emphasizes in a very forcible way the cultural decadence of the older seats 
of British learning in the period that extended from the generation of 
Newton and Bradley to the repeal of the Test Acts, which followed the 
foundation of new institutions such as those in which Davy,, Dalton, and 
Faraday carried out their work. It is instructive to note that some of the 
leaders of science like Priestley and Benjamin Franklin were intensely alive 



432 Science for the Citizen 

* 

to the political struggles which anticipated and accompanied the next stage 
of industrial expansion. Thus Smiles tells us: 

The impressionable mind of Dr. Priestley was moved in an extraordinary 
degree by the startling events which followed each other in quick succession 
at Paris; and he entered with zeal into the advocacy of the doctrines of liberty, 
equality, and fraternity, so vehemently promulgated by the French "friends 
of man." His chemical pursuits were for a time forgotten, and he wrote and 
preached of human brotherhood, and of the downfall of tyranny and priest- 
craft. He hailed with delight the successive acts of the National Assembly 
abolishing monarchy, nobility, church, corporations, and other long-established 
institutions. He had already been long and hotly engaged in polemical dis- 
cussions with the local clergy on disputed points of faith; and now he addressed 
a larger audience in a work which he published in answer to Mr. Burke's 
famous attack on the "French Revolution." Burke, in consequence, attacked 
him in the House of Commons; while the French Revolutionists, on the other 
hand, hailed him as a brother, and admitted him to the rights of French citizen- 
ship. These proceedings concentrated on Dr. Priestley an amount of local 
exasperation that shortly after burst forth in open outrage. On July 14, 1791, 
a public dinner was held at the principal hotel to celebrate the second anni- 
versary of the French Revolution. About eighty gentlemen were present but 
Priestley was not of the number. A mob collected outside, and after shouting 
"Church and King!" they proceeded to demolish the inn windows. The 
magistrates shut their eyes to the riotous proceedings, if they did not actually 
connive at them. A cry was raised, "To the New Meeting-house," the chapel 
in which Priestley ministered; and thither the mob surged. The door was at 
once burst open, and the place set on fire. . . . They made at once for Dr. 
Priestley's house at Fairhill, about a mile and a half distant. The Doctor and 
his family had escaped about half an hour before their arrival; and the house 
was at their mercy. They broke in at once, emptied the cellars, smashed the 
furniture, tore up the books in the library, destroyed the philosophical and 
chemical apparatus in the laboratory, and ended by setting fire to the house. 
The roads for miles around were afterwards found strewed with shreds of the 
valuable manuscripts in which were recorded the results of twenty years' labour 
and study a loss which Priestley continued bitterly to lament until the close 
of his life. . . . The members of the Lunar Society, or "the Lunatics," as 
they were popularly called, were especially marked for attack during the riots. 
. . . Boulton and Watt were not without apprehensions that an attack would 
be made upon them, as being the head and front of the "Philosophers" of 
Birmingham. They accordingly prepared for the worst; called their workmen 
together, pointed out to them the criminality of the rioters* proceedings, 
and placed arms in their hands on their promising to do their utmost to defend 
the premises if attacked. ... As for Dr. Priestley, he shook the dust of 
Birmingham from his feet, and fled to London; from thence emigrating to 
America, where he died in 1804. While such was the blind fury of the populace 
of Birmingham, the principles of the French Revolution found adherents in 
all parts of England. Clubs were formed in London and the principal pro- 
vincial towns, and a brisk correspondence was carried on between them and the 
Revolutionary leaders of France. Among those invested with the rights of 
French citizenship were Dr. Priestley, Mr. Wilberforce, Thomas Tooke, 
and Mr. (afterwards Sir) James Mackintosh. Thomas Paine and Dr. Priestley 
were chosen members of the National Convention; and though the former 
took his seat for Calais, the latter declined, on the ground of his inability 



The Rebirth of Materialism 433 

to speak the language sufficiently. Among those carried away by the political 
epidemic of the time were young James Watt and his friend Mr. Cooper of 
Manchester. In 1792 they were deputed, by the "Constitutional Society" of 
that town, to proceed to Paris and present an address of congratulation to the 
Jacobin Club, then known as the "Socit des Amis de la Constitution." While 
at Paris, young Watt seems to have taken an active part in the fiery agitation 
of the time. He was on intimate terms with the Jacobin leaders. Southey says 
that he was even the means of preventing a duel between Danton and Robes- 
pierre, to the former of whom he acted as second. Robespierre afterwards 
took occasion to denounce both Cooper and Watt as secret emissaries of Pitt, 
on which young Watt sprang into the tribune, pushing Robespierre aside, and 
defended himself in a strain of vehement eloquence which completely carried 
the assembly with him. 

Alger asserts that Watt junior was the anonymous Whig mentioned in 
Carlyle's narrative. The account of Priestley's career given in The Dictionary 
of National Biography says that Priestley himself was also 

elected a member for the Department of Orne in the National Convention. 
Other Departments followed suit, but while he accepted citizenship, he declined 
election. The majority of members of the Royal Society fought shy of him. 
Finding that they were rejecting candidates on political grounds, he withdrew 
attendance (1793). 

The mechanical innovations associated with what is usually called the 
Industrial Revolution have overshadowed an important feature of the change 
which manufacturing underwent during the latter half of the eighteenth 
century. It is mentioned in the concluding remarks of the following extract 
from Nef's Rise of the British Coal Industry: 

The expansion of industry and particularly the expansion of the woollen 
industry, diminished the space available for planting new trees. ... In all 
countries near the sea, writes an anonymous authority towards the end of 
Elizabeth's reign, "most of the woods are consumed and the ground converted 
to corn and pasture." ... If the growth of the woollen industry in particular 
discouraged the planting of trees, the demands of industry quickly drained 
the existing forests of their timber. For in that age wood was the raw material 
of all industry to an extent which it is difficult for us now to conceive. Charcoal 
had to be mixed with saltpetre in preparing gunpowder. From the bark of 
trees workmen extracted a sap then indispensable in making pitch and tar, 
with which to caulk the hulls of ships, and from wood ashes came potash, 
an essential constituent for the production of soap, glass and saltpetre. The 
principal drain caused by the expansion of industry arose . . . from the 
demands for it as building material and as fuel. Ours has often been called 
an age of coal and iron, and it is perhaps no less appropriate to call the sixteenth 
and seventeenth centuries an age of timber. ... It is unnecessary to dwell 
at length on the many thousand uses for firewood in early industry. It is sufficient 
to point out that no change could be wrought in ore or metal without the aid 
of fuel, that substantial quantities of wood and charcoal were being consumed 
in making starch, refining sugar, baking bread, firing pottery, tiles, bricks, and 
tobacco pipes, drying malt and hops, and boiling soap. . . . Every increase 
in the quantity of bricks, or saltpetre, lime or salt manufactured with wood 
fuel involved serious new encroachments upon native timber resources. But 



434 Science for the Citizen 

the chief wastage was caused by glass makers and smelters. . . . The smelting 
of iron was only a part of the larger problem of smelting and refining all metals, 
and that again was only a part of the still larger problem of making coal a suit- 
able substitute for wood in industry generally, and of reducing the total con- 
sumption of fuel of all kinds in a given process. . . . In 1623 a certain Lewyn 
van Hack had undertaken to extract silver from lead in Cardiganshire, using 
sea-coal in the smelting and charcoal "only in the refyning." We know nothing 
about his methods or his success, but in an article written 1678, and devoted 
to the methods of separating silver and other bodies from lead ore, Dr. Chris- 
topher Merret remarks that the "latest invention is a new furnace. The con- 
venience ... is, that a little fire, and that of New Castle coals, will do the 
work." ... In 1620 the Crown granted a patent for "charking earth fuel" 
to be used in smelting, and this is the first unmistakable reference to such an 
attempt that has been found. . . . With coke made in much the same way, 
Abraham Darby solved the problem of iron smelting, which, unlike the smelting 
of lead and tin, could not be accomplished with raw coal, even after the inven- 
tion of the reverberatory furnace. . . . The ultimate discovery by the elder 
Darby at the beginning of the eighteenth century of a successful process 
depended on innumerable experiments . . . often directed towards the 
smelting of metals other than iron, and even more often not directed towards a 
solution of the smelting problem at all. 

The activities of Roebuck, whose Carron Works at Stirling took a prom- 
inent part in the introduction of coal for reducing metal ores, illustrate the 
words italicized in the foregoing citation. While "the furnaces and forges 
had eaten up the woodlands of Sussex," and, as the Hammonds tell us, "were 
beginning to strip less promising districts bare," all the manufactures which 
depended on wood were forced into the search for material substitutes. One 
of the more important by-products was the alkaline potashes prepared by 
incinerating charcoal. Potashes were used for the making of glass and of 
soap, and for the cleaning of wool. Keir, whose house is mentioned by Smiles 
in a previous quotation, was one of the first to set up a factory for making 
alkali from sea salt. By the end of the century the commercial production 
of alkali was well established. 

The foundation of the synthetic alkali industry was made possible by 
the commercial production of sulphuric acid. Sulphuric acid was indeed the 
parent substance of modern chemical manufacture. Its composition will be 
dealt with more fully in the next chapter. Brimstone burns in air to form 
sulphur dioxide, a pungent gas which dissolves in water to form sulphurous 
acid. Sulphurous acid dissolves metallic oxides and alkalis forming salts called 
"sulphites." When heated with air in the presence of spongy platinum, 
sulphur dioxide is partially converted into sulphur trioxide, which contains 
more oxygen. Sulphur trioxide combines with water to form a heavy 
oily liquid, "vitriol" or sulphuric acid, which dissolves metals, alkalis, 
etc., thereby forming metallic "sulphates." The present world output of 
sulphuric acid is about 5,000,000 tons per year. Writing about 1840, a century 
after Ward's patent was first put into operation, Liebig said : 

We may judge with great accuracy the commercial prosperity of a country 
from the amount of sulphuric acid it consumes.* 

* JJebig's Letters on Chemistry, 



The Rebirth of Materialism 435 

The preceding remarks describe how sulphuric acid is made nowadays. The 
early history of sulphuric acid is briefly recounted by Lunge* as follows: 

Gerhard Dornaeus (1570) described its properties accurately; Libavius (1595) 
recognized the identity of the acids from different processes of preparation; 
the same was done by Angelus Sata (1613)., who pointed out the fact, which had 
sunk into oblivion since Basilius, that sulphuric acid can be obtained by 
burning sulphur in moist vessels (of course with access of air); after that time 
it was prepared by the apothecaries in this way. An essential improvement, 
viz. the addition of a little saltpetre, was introduced in 1666 by Nicolas le 
Fevre and Nicolas Lemery. ... A quack doctor of the name of Ward first 
carried on sulphuric-acid making on a large scale at Richmond near London, 
probably a little before 1740. Ward employed large glass vessels up to 66 
gallons capacity, which stood in two rows in a sand-bath, and which were 
provided with horizontally projecting necks; at the bottom they contained a 
little water. In each neck there was an earthenware pot, and on this a small 
red-hot iron dish, into which a mixture of one part saltpetre and eight parts 
of brimstone were put; then the neck of the bottle was closed with a wooden 
plug; on the combustion being finished, fresh air was allowed to enter the 
vessel, and the operation was repeated till the acid had become strong enough 
to pay for concentrating in glass retorts. . . . Ward's process, troublesome 
as it is, reduced the price of the acid from 2s. 6d. per ounce (the price of the 
acid from copperas or from burning brimstone under a moist glass jar) to 
2s. per Ib. 

Roebuck improved on the Ward process by introducing lead chambers 
which reduced the expense incurred by the use of large glass vessels. Prosser 
(Birmingham Inventors and Inventions) says that he set up a manufactory at 
Steelhouse Lane, Birmingham, with his partner, Mr. Samuel Garbett, in 
1746, and that the works, afterwards sold to Alston and Sons, continued to 
make the acid till 1825. Lunge tells us that soon after he set up his factory at 
Prestonpans several others were started in England : 

Soon other works followed at Bridgenorth, and at Dowles in Worcestershire 
where the chambers were already made 10 feet square; in 1772 there was a 
factory erected in London with 71 cylindrical lead chambers, each 6 feet 
diameter and 6 feet high. In 1797 there were already six or eight works in 
Glasgow alone. According to the statements given in Mactear's Report of the 
Alkali and Bleaching-Powder Manufacture in the Glasgow District (p. 8), the 
acid at that time cost the Glasgow manufacturers 32 per ton, and was sold 
at 54. At Radcliffe, near Manchester, it cost, in 1799, 21 10s. per ton, 
without interest on capital. 

According to Lunge the acid was first sold for the bleaching of linen. 
Since Home did not receive his medal till six years after the factory at 
Prestonpans began production, it seems unlikely that this was the original 
intention. Before making experiments on the lead-chamber process Roebuck 
had been engaged in work on refining precious metals, and it is possible that 
vitriol was used for dissolving traces of copper in the latter, or for making 

* Since this was written H. W. Dickinson, the biographer of Boulton, has published 
an account of the early history of sulphuric acid in the Journal of the Newcomen 
Society. 



436 Science for the Citizen 

nitric acid from saltpetre to dissolve out lead. We also learn* that Achard, 
a pioneer in the commercialization of beet sugar, used sulphuric acid (1792) 
for purification of beet juice. In England, Ward's product may have been 
wanted for the purification of clay used in the Potteries. The invention of the 
hydrogen balloon was an incidental consequence of its preparation in large 
quantities. 

In a small way it had been used by apothecaries for making the aperient 
called Glauber's salt. Glauber's salt, or sodium sulphate, was made by 
heating brine (the solid content of which is mainly sodium chloride) with 
vitriol. In the reaction which ensues a highly soluble pungent gas (now called 
hydrochloric acid) is evolved. The two products of this reaction were the 
parent substances for two other chemical industries established before the 
century ended. New methods for bleaching linen stimulated industrialists 
to seek for new expedients, and led to the introduction of chlorine gas, and 
bleaching powder prepared from it. The former is prepared by heating 
hydrochloric acid with black manganese dioxide, the latter by passing 
chlorine over dry lime. The antecedents of this new chemical industry are 
described by Smiles: 

Among Watt's numerous scientific correspondents was M. Berthollet, the 
eminent French chemist, who communicated to him the process he had dis- 
covered of bleaching by chlorine. Watt proceeded to test the value of the dis- 
covery by experiment, after which he recommended his father-in-law, Mi. 
Macgregor, of Glasgow, to make trial of it on a larger scale. This, however, was 
postponed until Watt himself could find time to superintend it in person. At 
the end of 1787 we find him on a visit to Glasgow for the purpose, and writing 
to Boulton that he is making ready for the trial. "I mean," he writes, "to try it 
tomorrow, though I am somewhat afraid to attack so fierce and strong a beast. 
There is almost no bearing the fumes of it. After all, it does not appear that it 
will prove a cheap way of bleaching and it weakens the goods more than could 
be wished, whatever good it may do in the way of expedition." The experi- 
ment succeeded, and we find Mr. Macgregor, in the following February, 
"engaged in whitening 1,500 yards of linen by the process." The discovery, 
not being protected by a patent, was immediately made use of by other firms; 
but the offensive odour of the chlorine was found exceedingly objectionable, 
until it was discovered that chlorine could be absorbed by slaked lime, the 
solution of which possessed great bleaching power, and this process in course 
of time superseded all the old methods of bleaching by chlorine. 

The commercialization of bleaching powder was successfully accomplished 
by Tennant of Glasgow in 1799. Meanwhile Glauber's salt had been put to 
a new use. The ashes left by incinerating charcoal (mostly potassium carbon- 
ate) or sea weeds (mostly sodium carbonate) had been used as the chief 
source of alkali in the glass, soap, and wool industries from time immemorial. 
In medieval times the forests of eastern Europe were responsible for a con- 
siderable export trade of "potashes" for the English wool industry. During the 
revolutionary wars the available supplies had run so short in France that the 
Academy was forced to offer a large prize for a suitable method of making 
alkalis from other sources. The prize was won by Leblanc in 1791. The 
* Private communication from Mr. J. L. Mackie. 



The Rebirth of Materialism 437 

Leblanc process for making sodium carbonate consists of three stages. 
First sodium chloride is obtained from sea water, of which it forms about 
75 per cent of the solid matter, by evaporation to the stage when the crystals 



Term&nL of 
1799 



CHLO&IKE 
BerthsUat , Watt , 
far ikacT 
about 178O 



(Sodium carbonate) 



leblvrtc, 1*791 
JSLS. KZLT 



Hydrochloric ac 
"marina" acid 




QLAUBZ&S SALT 
(soduzm 



prec Loiis 




idphuric add 

Ward process, Richmond J/4O 
ieaxi chamber process, 'Roebuck 8t ^* 




TIME CHART III 

Beginnings of Modern Chemical Industry. 

of sodium chloride separate out. The sodium chloride is next heated with 
sulphuric acid. Hydrochloric acid gas escapes, and can be collected by passing 
it into water. Sodium sulphate or Glauber's salt remains behind. In the 



438 Science for the Citizen 

final stage this is dried and cooked in a furnace with coke (carbon). The 
oxygen is thus removed from it. Carbon dioxide escapes, and sodium sul- 
phide remains behind. The latter is heated with chalk (calcium carbonate) 
giving calcium sulphide, which is insoluble in water, and sodium carbonate 
(washing soda), which is easily removed because of its great solubility. The 
Leblanc process produces hydrochloric acid as a main by-product and a 
large number of less important products, including carbon dioxide which is 
now used for charging soda-water siphons. 

Although the Leblanc process which superseded its predecessors was 
undoubtedly more efficient, it did not differ in principle from earlier and less 
extensive commercial undertakings in France and in Britain. About 1780 
the Scots chemist Keir, at one time a partner of Boulton, established a factory 
at Tipton for the production of alkali from sea salt. According to his bio- 
grapher the first stage was the production of sodium sulphate, as in the 
Leblanc process. Collison took out an English patent essentially similar to 
the Leblanc recipe in 1782. An earlier suggestion of Scheele (1776) to make 
sodium carbonate by heating sea salt with lead oxide and treating the solution 
with fixed air is not known to have been used. 

About this time Boulton's works sponsored the beginnings of one of the 
most powerful of all the chemical industries in the ensuing century. In 1785 
a continental chemist, Maestrichte, had shown that coal gas could be used to 
light a lecture room. A Frenchman named Le Loss used it to light his own 
house in Paris in 1786. Between 1792 and 1802 Murdock, a Scots workman 
in the employment of Boulton and Watt, carried out successful experiments 
which led to the installation of an equipment in the engineering works of 
Boulton, Watt and Co. at Soho, Birmingham. Prosser (Birmingham Inventors) 
says: 

Murdock is entitled to the sole credit of being the first to bring gas-light 
into the region of practice. Part of the Soho establishment was regularly lighted 
with gas in 1798. On the occasion of the rejoicings at the Peace of Amiens in 
1802 the front of the building was lighted by gas. After that time it spread 
rapidly to other parts of the country. In 1808 he read a paper before the Royal 
Society giving an account of his investigations on the subject, for which he 
received the Rumford Gold Medal. 

Some ten years after the installation of coal gas in Soho commercial 
production of matches from phosphorus began. The preparation of phos- 
phorus from excrement had been undertaken during the seventeenth 
century, when Boyle had toyed with the possibility of making matches. In 
the latter half of the eighteenth century Scheele (1771) discovered how to 
make it from bone ash, for which a commercial demand existed in the china 
industry. Thomas Frye took out a patent for the use of bone ash as an 
ingredient of pottery in the mid-eighteenth century. At the time of Scheele's 
discovery, "bone china" was being manufactured by Spode of the Stafford- 
shire Potteries. When the nineteenth century began, the trained chemist 
had become essential to industry, and the encouragement of chemical 
research to make the fullest use of every by-product was a matter of the 
utmost importance to the new manufacturing class. 



The Rebirth of Materialism 439 

THE STRUCTURE AND CLASSIFICATION OF MATERIALS 

Before it is possible to find laws which embody correct rules of conduct 
it is necessary to observe relevant facts with accuracy and classify them intelli- 
gently. Although comprehensive principles underlying the manufacture of 
new substances and the discovery of new sources of materials did not emerge 
during the century and a half which followed the issue of Glauber's Pros- 
perity of Germany, the accumulative results of the recognition of a third 
realm of matter, including a variety of substances with totally distinct 
characteristics, had provided the only foundation on which a useful body of 
theory could be built. Through the realization that the new g^ses were not 
various forms of "air," but individual substances with characteristics as 
distinct and constant as those of pure solids and liquids, a classification of 
substances based on the way in which they are built up, and hence on the 
sources from which they can be obtained, was gradually replacing that of 
the alchemists. 

Black's work (1754) on carbon dioxide and that of Cavendish (1766) on 
hydrogen had been followed by a plethora of discoveries. Compounds which 
had been dismissed as fumes and smells were systematically investigated. 
Besides those of the four elementary gases (oxygen, nitrogen, hydrogen, and 
chlorine) the characteristics of about a dozen other gaseous substances had 
been studied by the end of the eighties. Priestley described the properties of 
sulphur dioxide (1770), hydrochloric acid gas, nitrous and nitric oxide (1772), 
ammonia (1774), and sulphuretted hydrogen (1777). Scheele and Priestley 
discovered silicon tetrafluoride in 1776. Scheele, who had first made chlorine in 
1774, also made sulphur trioxide. Carbon monoxide was first studied by de 
Lassone in 1776. Methane and ethylene (two other constituents of the mixture 
that is called coal gas) were respectively investigated by Berthollet (1785) and 
by a group of Dutch chemists. 

By the end of the eighteenth century it was possible to distinguish between 
a group of simple substances or elements solids like sulphur, carbon (char- 
coal), phosphorus, and the metals, gases like oxygen, hydrogen, and nitrogen 
which combine to form compound substances with equally definite charac- 
teristics such as density, solubility, colour, odour, texture, power to conduct 
heat, and so forth. The class called compounds included first and foremost 
the oxides formed by combination with oxygen. One large class of oxides 
were all acidic., i.e. dissolving like the oxides of sulphur, carbon, nitrogen, 
and phosphorus, in water to form "acids." The definition of an acid retains 
to this day its early association with the dyeing of textiles. Acid solutions were 
recognized by their power to turn the blue litmus dye red. In contradistinction 
another class, the oxides of the metals, are distinguished as basic, being 
capable of combining with acids to form more complex bodies called salts. 
Some of these basic oxides, like quicklime, unite with water to form more 
complex substances called caustic alkalis containing hydrogen as well as 
oxygen and metal (e.g. slaked lime). Salts, which can also be formed by 
dissolving a metal in a strong acid like nitric acid, sulphuric acid, and 
hydrochloric acid, were being classified according to the acid or acidic 
oxide from which they were derived. Thus gypsum (calcium sulphate), blue 
vitriol (copper sulphate), green vitriol (iron sulphate), and Epsom salts 



440 Science for the Citizen 

(magnesium sulphate) are all now recognized as members of the class of 
sulphates (salts formed by dissolving a metal or its oxide in sulphuric acid); 
potashes (potassium carbonate), malachite (copper carbonate), and chalk 
(calcium carbonate) as carbonates (salts formed by the union of a metal 
oxide and carbon dioxide); and common salt (sodium chloride), calomel 
(mercury chloride), and horn silver (silver chloride) as chlorides. 

Substances of another class were known to be compounds because they 
could be broken down into two or more other distinct substances. Not all 
substances recognized as elements or compounds a century and a half ago 
are regarded as such today. Some substances which we can now build up 
from their elements were still believed to be elements. The yellow gas 
chlorine, which is an element given off when manganese dioxide is heated 
with a solution of hydrochloric acid, was then believed to be an oxide. 
Chlorine unites with hydrogen in sunlight to form the colourless hydro- 
chloric acid gas. Since it was thought that all acids are formed from oxides, 
like carbon dioxide, sulphur dioxide, or phosphorus pentoxide, hydrochloric 
acid was at first supposed to be the oxide of an undiscovered element, and the 
use of manganese dioxide in its preparation seemed to support this conclusion. 
Repeated attempts to break it down eventually discouraged this belief. 

The new method of classification was not merely a question of what 
names we give to substances. It meant far more. Calling three substances 
litharge, galena, and sugar of lead, communicates nothing about the way 
they are built up. When we call them by their new names, lead oxide, lead 
sulphide, and lead acetate, we convey new information about the way in 
which they may be obtained, or the use to which they may be put. For in- 
stance, each is a source of lead. Most sulphides can be converted into oxides by 
heating with an oxidizing agent something that gives up oxygen readily. 
Thus the malodorous gas sulphuretted hydrogen or hydrogen sulphide, 
liberated by strong acids from metallic sulphides like pyrites, will burn in 
air to form water (hydrogen oxide) and sulphur dioxide. Most oxides of 
metals can be converted into metal by heating them with a "reducing" agent, 
such as hydrogen gas, which combines with the oxygen to form water vapour, 
or charcoal, which combines with the oxygen to form carbon dioxide. Among 
the best oxidizing agents for soluble substances are hydrogen peroxide and 
potassium permanganate. Among the best reducing agents are the bubbles 
of hydrogen given off when tin or zinc dust are dissolved in a strong acid 
like hydrochloric acid. Metallic acetates in general are converted into oxides 
by heat, and made by dissolving a metallic oxide or carbonate in acetic acid, 
which gives vinegar its sour taste. From such general knowledge of the 
characteristics of acetates, oxides, and sulphides, and general knowledge of 
the characteristics of lead compounds, we can condense a mass of useful 
information in the statement that a substance is the acetate, oxide, or sulphide 
of lead. 

Hence the new label for galena tells us something about how we should 
set about looking for the substance. Metallic sulphides give off sulphuretted 
hydrogen when treated with a strong mineral acid like hydrochloric. Lead 
salts are identified by the fact that nearly all chlorides are soluble in cold 
water, the notable exceptions being lead, silver, and mercury, of which 



The Rebirth of Materialism 441 

the last two are not soluble in hot water. So if a lead compound is dissolved 
in nitric acid, all the salts of which (nitrates) are soluble, a precipitate which 
redissolves in hot water is formed on adding hydrochloric acid. 

In short, the new method of classification based on the way things are 
built up is an inventory of man's resources of materials, known and as yet 
unknown. The discovery of the metal chromium which is now used as an 
ingredient of rustless steel in the middle of the eighteenth century illustrates 
how the new method helps to discover new kinds of materials, as well as 
directing our attention to new sources of known materials. In the middle of 
the eighteenth century a Siberian mineral (now known to be lead chromate) 
was found to be the representative of a class of salts which did not corre- 
spond to those formed from the oxides of any known element. Subsequent 
search led to the discovery of a new metallic element. 

At the end of the eighteenth century the four elements of Aristotle had 
been replaced by about fifteen. Today we recognize about ninety. More than 
half of those known at the time of Lavoisier had been objects of use for 
several millennia. Those known to the ancients in a comparatively pure form 
included carbon, sulphur, and the metals gold, silver, iron, copper, tin, 
lead, antimony, mercury, and zinc. During the seventeenth and eighteenth 
centuries the list of solid elements was extended by the addition of phosphorus 
and several new metals, notably platinum, chromium, arsenic, and manganese. 
Three gaseous elements, oxygen, nitrogen, and hydrogen, which were dis- 
covered in the eighteenth century, had not been previously known as separate 
substances. Chlorine had been isolated, though not as yet known to be an 
element. 

The list of elements then known excluded many names which are familiar 
to any civilized person today. The commonest metallic elements were only 
known in their compounds. Sodium, whose chloride is the chief solid con- 
stituent of the sea common salt; potassium, whose nitrate is the essential 
constituent of gunpowder; calcium, whose carbonate (chalk) forms the cliifs 
of our shores, and whose oxide (lime) had been used from time immemorial 
for curing leather and making cement, were not known in the pure state. 
Boron, whose compounds were already used in medicine, had not been 
separated. Beryllium, a metal like magnesium, forming three crystalline 
salts which are the familiar jewels beryl, aquamarine, and emerald, was 
not known. 

To this list of common elements, unknown in Lavoisier's time, we must 
also add the two elements aluminium and silicon. With oxygen they form the 
overwhelming bulk of the superficial layers of the earth's crust. Aluminium, 
a metal, is now used in paint, kitchen utensils, piston heads, and aeroplane 
engines. Its oxide, tinted with various impurities, is the main constituent 
of the gems sapphire, ruby, and amethyst. In combination with silica 
(aluminium silicate) it is the chief constituent of common clay. It is ubiquitous, 
and vastly more abundant than any other metal to a depth of half a mile 
of the earth's crust; and even to a depth of 25 miles approaching the iron 
core of the earth it is at least seven times as abundant as iron. Silicon is not 
a metal. Its oxide (silica) is rock crystal or quartz. Sand is mainly silica 
with various impurities. Glass, which was originally made by heating sand 



442 Science for the Citizen 

with an alkali (lime or potashes) or basic oxide (e.g. litharge), is a mixture of 
various metallic silicates (commonly calcium, sodium, potassium, and lead). 
Window and table glass is a sodium calcium silicate mixture, flint glass and 
cut glass a lead potassium silicate mixture. Jena glass is a potassium calcium 
silicate mixture. 

The elements known in the eighteenth century did not include the com- 
paratively rare element radium, nor selenium, whose great sensitivity to light 
is used in television. The three elements (bromine, iodine, and fluorine) 
whose compounds closely resemble those of chlorine were not yet isolated. 
A silver compound of one of these (silver bromide), which is very sensitive 
to light, is the basis of the photographic industry. The solution of iodine 
(generally dissolved in a solution of potassium iodide) is a familiar antiseptic. 
Fluorine in combination with silica forms complex salts (fluosilicates) which 
occur as minerals. It is also a constituent of the acid called hydrofluoric acid 
which attacks glass and for this reason is used for etching. 



QUANTITATIVE ANALYSIS 

It is a common delusion to think that the introduction of a little mathe- 
matics into a subject necessarily makes it more scientific. Mathematics is 
useful when we wish to recognize the significance of measurements. In the 
early stages of a science the important thing is to know very thoroughly the 
characteristics of the things which we are studying. We can only make useful 
measurements when we are clear about what we are measuring. This was the 
task of chemistry in the eighteenth century. Till the beginning of the nine- 
teenth century, only one important rule of measurement in chemical reac- 
tions had been discovered. This is usually called the Law of Constant Pro- 
portions. It would be foolish to put a date to its discovery, because it represents 
a conclusion which gained increasing confidence as chemists acquired the 
habit of weighing all the ingredients and products of a process. 

The Law of Constant Proportions is that the ratio of the weights of 
the ingredients used up in making the same substance is always the 
same. If the ingredients are themselves elements this is the same as saying 
that the proportions by weight of the constituent elements in any compound 
is fixed. To prove this rule it is necessary to weigh separately each constituent, 
as Lavoisier did in his experiments on calcination. Once we are satisfied that 
it is true much labour may be saved. We can use it to calculate some quantities 
if we have recorded others. This may be illustrated by the way in which we 
can find the proportion by weight of magnesium and oxygen in the oxide 
of the metal (Fig. 252). A tube with a wire to hold it in place on the balance 
and a plug of glass wool are first weighed. A strip of magnesium is then put 
in the tube and the glass wool is inserted. The glass wool and tube with 
magnesium in it are now weighed, and the difference between the two 
weighings is the weight of magnesium added. If the tube is now held in a 
flame, the magnesium ignites with the blinding brilliancy now familiar in 
flashlight photography, and a dense cloud of white powder (magnesium 
oxide) is formed. The glass wool prevents explosion by admitting air freely, 
while at the same time stopping the particles of oxide from getting out. A 



The Rebirth of Materialism 443 

third weighing when the tube has cooled now shows an increase in weight. 
If we call the weight of the tube and glass wool before adding magnesium a, 
the glass wool and tube with magnesium b, and the same after ignition c, the 
weight of magnesium is (b a), the weight of magnesium oxide formed is 
(c a), and the weight of oxygen with which (b a) grams of magnesium 
combines is (c a) (b a) = (c b). Providing the magnesium is as 
pure as we can get it, the ratio of (b a) magnesium to (c b) oxygen is 
always the same. 

How we might determine the proportion by weight of silver and chlorine 
in silver chloride provides a second illustration of what chemists call gravi- 
metric analysis. A clean dry centrifuge tube is weighed. A small quantity of 
pure silver is placed in it, and it is weighed again to get the amount of silver 
(b a). A solution of moderately strong nitric acid is added to dissolve 
the silver. From the solution of silver nitrate so formed the insoluble chloride 




FIG. 252. GRAVIMETRIC ANALYSIS OF MAGNESIUM OXIDE 

is precipitated, if a strong solution of pure sodium chloride is added. This 
white powder (which darkens in sunlight) is thrown to the bottom of the tube 
in a hard cake, after rotation in a high-speed centrifuge. The fluid can now 
be poured off, the tube with the cake of silver chloride rinsed with pure 
water and placed in a warm oven till it is dry, i.e. when repeated weighing 
gives constant results. The difference between the final weighing (c) of the 
tube plus silver chloride and the second weighing (b) of the tube plus silver 
is the weight of chlorine which unites with the weight (b a) of silver. 

In both these examples the irksome necessity of weighing a gas is avoided 
by using the Law of Conservation (p. 393) to get the result indirectly. In all 
analysis one or other of the constituents is obtained by inference. One useful 
method of analysis avoids repeated weighing and drying by measuring 
volumes of solutions of known strength. Standard solutions for "volumetric" 
analysis are made in large quantities by weighing out accurately a suitable 
amount of a reagent and dissolving it in a "volumetric" flask (Fig. 253). 
This bears a mark corresponding to the level of water which exactly occupies 
a stated volume, e.g. 1,000 cubic centimetres (1 litre) at a stated temperature. 

Let us suppose that we wish to know in what proportions by weight sodium 



444 



Science for the Citizen 



and chlorine unite to form sodium chloride (common salt), and that we already 
know how much silver chloride is formed from a given quantity of silver 
nitrate. We have to start with a solution of silver nitrate formed by dissolving 
exactly x grams in a litre, so that 1 c.c. of the solution contains 0-OOlx grams. 
If y grams of silver chloride are formed from x grams of silver nitrate, 1 c.c. of 
the solution is equivalent to 0-OOly grams of silver chloride, and if y grams of 
silver chloride contains z grams of chlorine, 1 c.c. of the solution is capable of 



Stepper r 




FIG. 253 

One litre volumetric flask (A), 20 c.c. pipette (B), and burette (C), used in volumetric 
analysis. 

throwing down O-OOl^ grams of chlorine as silver chloride precipitate. Silver 
nitrate is a common laboratory reagent for estimating chlorides, so a standard 
solution of it is usually kept in a laboratory. To find the proportions of sodium 
and chlorine in a solution of sodium chloride without going to the trouble 
of preparing and keeping the metal pure, we might make up a standard 
solution of sodium chloride by dissolving n grams of sodium chloride in a 
litre, so that 1 c.c. of the solution contained 0-OOlw grams. We now measure 
out a fixed quantity, e.g. 10 c.c., of this into a glass vessel with a marked tube 
("pipette"). This is done by sucking up the fluid with the mouth and keeping 
the level exactly at the mark by applying a finger to the orifice. Finally, we 
drop into this vessel the standard solution of silver nitrate from a graduated 



The Rebirth of Materialism 445 

tube (burette) with a tap which can be turned off when all the sodium chloride 
has been used up to f